The claim vastly depends on what you consider parts... and even on the model of car in question. Given that Tesla S used AA-sized cells for its battery pack... how many of those did it take to power the car? (I bet more than 200.)
Straubel pointed to the wide variety of lithium-ion battery cells—the parts of a battery pack that actually store energy—that the company is testing. This included a row of small cylindrical cells about the size of AA batteries—the kind Tesla uses in the Model S.
By choosing smaller, cylindrical cells, Tesla saved on manufacturing costs—their costs have been driven down by economies of scale for the laptop industry, for which the cells were developed.
Count them in this photo if you want (source):

(The source actually did that for us: 444 or 384 cells depending on pack model: 85kWh or 60kWh.)
The same is true for Tesla Model 3 basically, although a slightly thicker and taller (custom) cell dubbed 2170 is used instead of the industry statndard 18650.
The advantage claim is a bit more believable for moving parts (as covered in the other answer).
Also Toyota says
A single car has about 30,000 parts, counting every part down to the smallest screws.
So while one may make valid claims about the superiority of electric cars in various ways, the comparison quoted in the question is just not believable to me.
Likewise, an (academic) source says:
A typical car requires 20,000 to 30,000 parts, each usually manufactured in a different facility and by different company groups.
HBR does have slightly different view:
The typical car contains about 2,000 functional components, 30,000 parts, and 10 million lines of software code.
So maybe the OP's page is talking about these "functional components" as opposed to parts... but from that HBR page I can't tell how these are defined or counted.
I don't know if any car-industry standards exist for counting "functional components", but a textbook on such decompositions in general (using cars as example), essentially says the number is arbitrary and not related to manufacturing:

Figure 7.4 exhibits the black-box model of a car. It corresponds with the
driver’s perspective on a car. So, the driver is the using system and the car
is the used system. Through changing the values of the input variables
(e.g., the position of the steering wheel) the driver is able to change the
values of the output variables (e.g., the direction of the car). Ideally, as we
have seen, the (transfer) function is a mathematical relationship between
the input variables and the output variables. However, for most concrete
systems, e.g. for cars, this is hardly possible. Therefore, in practice, the notion of function is a rather informal, loosely defined notion of what kinds
of (functional) behavior can be caused through manipulating the input
variables. In this respect one also often uses the term “functionality”, a
term that we prefer to “function”.
If the transfer function is too complicated to understand, the technique of
functional decomposition can be applied, through which the black-box
model of a system is replaced by a structure of submodels of with more
readily understandable functions. Figure 7.4 shows a possible decomposition
of a car. If a component is still too complicated, it can be further decomposed.
The exhibited decomposition of a car could be very helpful for
a driving instructor to explain its functionality to a new student. Note,
however, that the knowledge that one acquires about a system by means of
functional decomposition is only functional knowledge, nothing less and
nothing more. The only thing one does in functionally decomposing is help
explain the functionality of a system, as we showed for the car example.
The idea that functional decomposition ultimately leads to knowing the
construction of the system is a widespread misunderstanding.
To elaborate
this, a BB model is a purely conceptual division of the function or functionality
of a system, independent of its construction and operation. Therefore,
one can make virtually any decomposition one likes, and one can
freely add or remove functional components.
(From J. Dietz Enterprise Ontology: Theory and Methodology, Springer, 2006).
Britannica has a similar presentation

The stuff that ventsyv is talking about is based on these aggregate functional notions rather than actual part counts. In this view a battery is one thing regardless how many cells it has etc.
Finally, does any of this overall part (or subsystem) counting matter in economic terms? I think the answer is no, because a 95-page UBS comparison of internal combustion and electric vehicles never mentions total (or component/subsystem) part count as an economic advantage (either way). It does mention moving/powertrain parts though (extensively):
Mechanical complexity is much lower [in EVs], whereas electronic complexity is higher. We counted 24 moving parts in the Bolt's powertrain, versus 149 in the Golf. The powertrain electronics content is $4k higher on the tier-1 level, motor included.

And it also mentions spare/replaceable parts being fewer in EV's:
Also, revenues from the lucrative spare parts business, which accounts for ~20% of EBIT, are likely to drop by ~60% in the long term in an EV world. However, this scenario is several decades away.
And again (later):
A separate point not to be ignored: Because EVs have much fewer moving and
wearing parts, the attractive spare parts business, which represents ~10-15% of an
OEM's EBIT, is likely to shrink considerably long-term. However, this should take
another 15-20 years longer, due to the replacement cycle of the existing car parc
and one more time
The Bolt is almost maintenance-free. Not only do fewer parts
need to be replaced over the car's life, it also does not require a regular change
of fluids, such as engine oil. On our analysis, the after-sales revenue pool could
drop by ~60% or >$400 per vehicle per year. This should pose a major
challenge for dealerships, which typically generate >40% of their gross profit
pool in service and maintenance.

But the overall part (or component/system) count at assembly seems irrelevant, at least at this depth of analysis... which is still quite substantial.
The power train is considered cheaper overall in EVs (this is currently offset by the battery in the figure 6 above).

The Bolt's powertrain is much simpler
than the Golf's from a mechanical point of view:
The e-motor itself is much less complex than the combustion engine. Bearings
aside, there are only three moving parts. Modern e-motors are brushless, ie,
maintenance-free. The Golf's 4-cylinder engine has 113 moving parts. On top,
spark plugs need to be replaced and engine oil needs to be changed regularly.
The combustion engine has a limited usable rotation range, between c800-
6,000 rpm. Also, its torque is not constant over the usable rpm range (unlike
the e-motor). Therefore, a complex gearbox and clutch (or torque converter)
are needed. The Golf's 6-speed automatic transmission has 27 moving parts.
Gearboxes and clutches also wear. After mileage of 150k kilometres, gearbox
replacements begin to rise significantly. In contrast, the Bolt has a very simple
single-speed gearbox with only four gear wheels. We expect no maintenance
or replacement to be required over the life of the car.
This is of course translates into an assembly advantage for the power train:
The motor in a BEV replaces the engine and transmission in an ICE vehicle and will
contain a significantly lower number of moving parts. For instance, we expect that
electric vehicles will have 6-7 bearings in the drive module (e-motor and mini
gearbox) compared to 40-50 bearings in a traditional ICE. [...] We also expect significantly less machining will be
required for the e-powertrain vs. conventional ICEs. Our channel checks indicate
up to 80% of the cutting tool work needed to manufacture a car happens in the
combustion engine. Significantly less machining is required for the e-motor.
The latter is because electric motors' rotor and stator are made of (electrically isolated) stamped laminates (making them of one block is a no-no because of Eddy current losses). They also talk extensively about the battery in EVs and what makes it expensive. They do note that currently there's a substantial overhead of assembling the Bolt battery pack (in Bolt is made of 288 pouches):
Finally, economies of scale and the learning curve in cell and pack
assembly should bring further savings. In today's $3,600 pack mark-up, only
~25% relates to materials used. This points to high fixed costs in a sub-scale
production environment. We assume a contribution from economies of scale
of $10/kWh.
But with all this detailed analysis, it never gets to overall part (or system) count being an economic advantage. They only say that
General assembly is a similar process for BEVs and ICE vehicles.