Hard legal question to answer, but I would say in would fall on YouTube as long as you did NOT "steal" it from the web page in a way that is not the normal copy/paste of the given HTML code sometimes supplied by YouTube. I am not a lawyer, so you would be better to seek real legal advice if you suspect you may have a problem in this area.
I've had people complain about videos we embed, and once I point out that their beef is with YouTube, they understand that they have to kill it at the source. I think it would be a huge stretch to find you liable for a video that came with embedding enabled from YouTube unless you were doing something willfully malicious.
Copyright owners of audiovisual works have exclusive rights such as distributing, publicly performing, and publicly displaying the works. A person who violates these rights is infringing. These exclusive rights do though, have a few limitations, such as for Fair use and Reproduction by libraries and archives.