my favourite movie when i was a child was:
...but it's not
something i'd recommend you to show your children
(i've watched it several times, before the family realised it and prevented me from watching it again)...
it's not exactly a pirate movie, but the following scene seemed to me like a tribute to the MI (coulda been just a coincidence, but i doubt it):
the wig and the monkeys look like the reference to the MI.
you can watch a scene from it here
(the scene length is longer, and the quality's better).