Doctor Who has always been funny. After all, when you have a character that travels throughout time and space in a police phone box to battle mutant alien fascists that use plungers as a weapon, you have to be a little tongue-in-cheek. Even though some of Doctor Who's most hilarious moments reside in the show's first run on television (including Tom Baker's memorable quip, "you're a beautiful woman probably.") the current version of Doctor Who is designed to be infinitely more comedic than its predecessor. In truth, before the 2005 reboot, while most Whovians were laughing with Doctor Who, the broader culture laughed at the series.
However, even though pop culture wasn't as kind to Doctor Who in the 20th century as it has become in the 21st, there were always a few sketches and spoofs that lovingly poked fun at the series. In the video linked above, you can check out a great spoof from the 80s that captures the tone of the first iteration of the series.