I was listening to “I Dig Rock and Roll Music” and enjoyed Peter, Paul and Mary’s impressions/imitations of The Mamas and The Papas, Donovan and The Beatles. It brought to mind the version of Simple Twist of Fate where Joan Baez imitates Dylan for a verse.
Others?
The Bangles’ “Gliiter Years” has a Bowie imitation at about 2:35.
Mika’s Grace Kelly has him impersonating Freddie Mercury.
Frank Zappa’s Flakes has a funny Bob Dylan parody section, done by Adrian Belew, I think.
I don’t know if instrumental impressions count, but there’s a great song named Guitar Heroes on Richard Thompson’s latest album Still where he plays in the style of Django Reinhardt, Les Paul, Chuck Berry, James Burton, Hank Marvin, and, well, Richard Thompson.
“Rio” by [del]Duran Duran[/del] Goldfinger – a punked-up cover song which, around 1:55, inexplicably transforms into a Ronnie James Dio tribute.
The one that springs to mind is Man In The Moon by** R.E.M**. where the line ‘Andy are you goofing on Elvis’ is followed by an Elvis style ‘Hey Baby’
TCMF-2L
“Man on the Moon” by R.E.M includes a line where Michael Stipe mimics Elvis.
A Simple Desultory Philippic by Simon & Garfunkel, with Simon impersonating Dylan.
Prince, on Sign o’ The Times, on the song The Ballad of Dorothy Parker: he asks her for her favorite song and she says Joni singing “Help me, I think I’m fallin’…” brrrring! phone rings…
He and his backing vocalists do a nice rendition of the Help Me line like Joni Mitchell.
Great song.
The Led Zeppelin song “The Crunge” has an imitation of James Brown at the end (“let’s take it to the bridge”), although you could argue that the whole song is an attempt at Brown’s style, which would disqualify it for this thread.
Motorhead’s R.A.M.O.N.E.S. is intended to sound like the Ramones.
The lead singer of Dread Zeppelin is an Elvis impersonator.
On the song “Time Stands Still”, Gary Lewis does an impression of his father on the final verse (around 1:20).
There’s little point in mentioning Weird Al in general – but when he’s doing his take on Lady Gaga in particular, he ribs her by having a brief Madonna impression come in.
The whole Deface The Music album by Todd Rundgren and Utopia is a homage to The Beatles.
Led Zeppelins Bring it on home begins with a letter perfect imitation of Sonny Boy Williamson.
David Allen Coe does a pretty good imitation of Waylon Jennings, Charlie Pride and Merle Haggard in “You Never Even Called Me by My Name”. And it calls out to “Hello Walls” in the second verse.
And of course the Beatles managed to do a pretty good impression of the Rutles… 
I don’t know whether songs by one artist that deliberately sound like they’re by a different artist is within the scope of what the OP is looking for, but if so, I’ve always thought that Adrian Belew’s “Big Blue Sun” was a great ELO song.
I always thought Hide your love away by the Beatles was Dillonesque.
Also Queen’s Crazy Little Thing Called Love has hips like Elvis
And Wild Horses by the Rolling Stones isn’t a specific band/singer, but a foray into country twang from their usual fare
Todds LP Faithful from 1975 has a whole side of recreations of 60s classics, including all vocal mannerisms. Beatles, Beach Boys, yardbirds, Hendrix etc.
The other side is one of the best sides of original rock I have ever heard.