Honestly, as far as I can tell, there doesn't seem to be a specific widely known name for this kind of folding canopy mechanism. That's not to say that it doesn't have a name, but that all of the names and terms I've found are either so generic that they could apply to many different kinds of folding mechanisms, so obscure that they're likely to draw blank stares from most people (and few search results from Google) even if they're familiar with the mechanism, or they have multiple meanings that can confuse people (and Google) if you use them without additional context.
That said, if you really want a name for it, I do have one for you:
It's a calash top.
Specifically, the term calash (also spelled caleche or calèche) can refer to:
- a type of horse-drawn carriage (also known as a barouche) with a soft folding hood over the rear seats, using a folding mechanism similar to what you depict;
- a calash top, the folding top of the aforementioned carriage, or one similar to it; or
- a calash bonnet, a type of 18th century women's headgear resembling the hood of the calash carriage, and also using a similar (although simpler) folding mechanism.
Pictures: 1) A calash or barouche carriage, showing the distinctive folding top. Photo taken by Wikimedia Commons user Daderot in the National Museum of American History, Washington, DC, and released to the public domain via the CC-Zero 1.0 public domain dedication. 2) A green silk taffeta calash bonnet. Photo courtesy of the Missouri Historical Society, St. Louis, via Wikimedia Commons.
If you do a Google image search for calash, you'll probably get about an even mix of the headgear and the carriages. And, of course, a bunch of random results like hotels and companies and health supplements named "Calash", as well as a mix of other types of horse-drawn carriages that may or may not be properly called calashes, despite not having the distinctive calash top. A general web search first brings up a bunch of dictionary definitions, followed by mostly of random pages.
Wikipedia has an article for the carriage (under barouche), but apparently none for the headgear yet, although it is mentioned in the general article on bonnets. At least Wikimedia Commons has pictures of both the carriages and the headgear.
All in all, it does not seem to be a widely used term at all. But it's probably the best name there is for what you're looking for.
At least, for what it's worth, if you search Google for "calash" baby cart (with quotes to force the word "calash" to be included in the results), you'll get a bunch of images showing baby carriages with similar hoods as in your question. Then again, searching e.g. for convertible baby cart also brings up a very similar set of images. Some sites even claim that "calash" can be a generic synonym for a baby carriage, or at least for the kind with a folding top. So take it for what it's worth.