It seems that the answer changes with time.
To look at the meta of what is going on, it seems that there are two generalities to the desired effect.
1) You can try and 'hack' the iframe code itself.
2) You can try and create a container to trick the iframe into thinking it should display HD.
Let's do both.
[ SPECIFIC IFRAME CODE]
You can possibly edit the typical embed youtube iframe link using current standards. I would recommend using a base size that would demand that size anyways and proceeding with step two to resize it.
Look up a current listing such as the one on h3xed to see the way youtube calls the files when embedded.
Of note, I didn't find the following code anywhere, I discovered it. I need to call videos that are 720. I was looking for the answer to this question and when viewing the file I noted that it said 720p60 as the actual youtube setting.
So I altered what seems to have worked before and sure enough...
<div class="responsive-container" >
<iframe width="780" height="480"
src="https://www.youtube.com/embed/DFzUdTUaAr4?rel=0&vq=hd720p60" frameborder="0" allowfullscreen></iframe>
</div>
worked.
Note that essentially I added ?rel=0&vq=hd720p60
And made the iframe size large enough to demand hd.
[ CREATING A CONTAINER ]
This works because you are asking youtube for a higher quality video and then going behind it's back and resizing it to fit the space you desire. Although you directly ask how to embed, I'm assuming you're asking to embed whenever and wherever you want - not being restricted to giant videos on page for high quality files.
A simple responsive container works well since iframes are made to be controlled through CSS. Using code similar to that found on thenewcode's Force-Embedded-Youtube-Videos-To-Play-In-HD article we create a code that restricts the aspect ratio to a limited size.
.responsive-container {
position: relative;
padding-bottom: 53.25%;
padding-top: 30px;
height: 0;
overflow: hidden;
}
.responsive-container,
.responsive-container iframe {
max-width: 1280px;
max-height: 720px;
}
.responsive-container iframe {
position: absolute;
top: 0; left: 0;
width: 100%;
height: 100%;
}
*Of note: 'Legacy' code of &fmt=35, &fmt=22, or &fmt=37
works at this point for video links. The youtube video opens up at this specific quality.
Also note that you also have to notice a difference in tdl between youtube videos and embedded videos. They are (from my experience) not cross compatible. * (youtube.com/embeded... VS youtu.be/...)