It's possible they are actually talking about the rate of energy given off by the Sun that reaches the Earth. However, they are not talking about the amount of energy released, but rather the rate of energy released.
The "laser at Magurele, Romania" is actually part of the Extreme Light Infrastructure, a pan-European research project, described by Wikipedia as
...a laser facility that aims to host the most intense beamline system worldwide, develop new interdisciplinary research opportunities with light from these lasers and secondary radiation derived from them, and make them available to an international scientific user community.
According to the Wikipedia article, on 13 March 2019, the ELI NP Research Centre, which is the facility located in Magurele, released a communication regarding the results of a demonstration test.
On March 13, 2019, Magurele held the public communication of the ELI-NP high-power laser system test results, which was also a demonstration test, confirming the achievement of the power of 10 [Petawatts].
A Petawatt is the equivalent of 1,000,000,000,000,000 (15 zeroes), or 10^15 Watts, as the prefix Peta describes. Therefore, a 10 Petawatt laser would be a 10x10^15, or 10^16 Watts.
Per this report from Sandia National Laboratories, they calculate the amount of solar power that reaches the earth's surface as 89,300 Terawatts.
A Terawatt is the equivalent of 1,000,000,000,000 (12 zeroes) or 10^12 watts. A Petawatt is equal to 1,000 Terawatts, so you can easily convert between the two by dividing the number of Terawatts by 1,000 to get the number of Petawatts. Therefore, the amount of solar power hitting the Earth in Petawatts is 89.3 Petawatts.
Dividing the output of the ELI-NP laser test by the energy output of the Sun that reaches the surface of the Earth results in
10 Petawatts / 89.3 Petawatts = 11.198%
which is approximately 10%. Note however, that this does not mean that the laser is continuously generating 10% of the sun's energy. Per the Wikipedia article on Watt
[The Watt] is defined as a derived unit of 1 joule per second,1 and is used to quantify the rate of energy transfer.
Further down, the page has a section on the distinction between "Watts" and "Watt-hours".
The terms power and energy are frequently confused. Power is the rate at which energy is generated or consumed and hence is measured in units (e.g. watts) that represent energy per unit time.
For example, when a light bulb with a power rating of 100W is turned on for one hour, the energy used is 100 watt hours (W·h), 0.1 kilowatt hour, or 360 kJ. This same amount of energy would light a 40-watt bulb for 2.5 hours, or a 50-watt bulb for 2 hours.
Power stations are rated using units of power, typically megawatts or gigawatts (for example, the Three Gorges Dam is rated at approximately 22 gigawatts). This reflects the maximum power output it can achieve at any point in time. A power station's annual energy output, however, would be recorded using units of energy (not power), typically gigawatt hours. Major energy production or consumption is often expressed as terawatt hours for a given period; often a calendar year or financial year. One terawatt hour of energy is equal to a sustained power delivery of one terawatt for one hour, or approximately 114 megawatts for a period of one year.
Typically, these kinds of experimental lasers are not constantly on, and fire for an extremely short period of time. Per the article on the National Ignition Facility, a facility with a similar, albeit less powerful laser
NIF aims to create a single 500 terawatt (TW) peak flash of light that reaches the target from numerous directions at the same time, within a few picoseconds.
A 10 Petawatt laser fired for a single picosecond would consume
10^16 Watts * (1 / 10^12) seconds = 10,000 Watt-seconds
10,000 Watt-seconds / 3,600 (seconds/hour) = 2.78 Watt-hours
Compare this number to the energy consumption of the world which was approximately 22 Terawatt-hours in 2017, and the amount of energy consumed by this laser is completely insignificant, accounting for less than a trillionth of the world's energy consumption.