While there is no Dr. Who on YouTube, you can watch many episodes on Hulu. Even some of the very first ones can be viewed. As an ex-cable viewer, I have found Hulu to be useful in "cutting the cable," even more now that one can bypass most commercials.
This comedy parody of Dr. Who is available on YouTube, and if you have some familiarity with the series you will find it amusing, modestly amusing anyway. Rowan Atkinson does a good job on the parody, with a cameo by Hugh Grant.