It depends. If the whole band is trying to stay true to the cover, then I'd say play it like the cover. That said, there would never be remakes or Weird Al, if everything was played like the original.
In the case of the Al parodies, the recordings are 99.9% -
or better - faithful to the original artist's recording, both in parts, sounds, and production. Granted, in the early days we weren't quite that close, but we really started to shine by the early '90s.
Apart from the gig with Al, I play covers as close as humanly possible (that is, without reading the parts...) for a couple of reasons.
The original drum parts often account for the groove and feel of the song. Unless doing a radically different treatment, such as playing Day Tripper inna reggae style, why change the feel? Why change the parts that make the song worth playing?
I have no personal drumming agenda, so I don't feel an urge to put my 'stamp' on covers. Honestly, I'm not going to improve on the original drummers' parts anyway. And it's fun to 'be' a bunch if drummers! Why be me? How boring.
But I do have a few deliberate twists on a classic Beatles' song. Every time I play "I Saw Her Standing There" I change the fills coming out of the bridges. Now, they're both Ringo fills, but from other songs. And the overall groove was nabbed from a McCartney performance of the song. Still a Ringo part, just not from that song. But it would take a real fanatic to figure out that I'm not playing 100% to the original, yet it sounds correct because everything I'm doing is Ringo and from the era.
Bermuda