bit obscure unless you’re steeped in the metal world, but “Bye Bye Beautiful” by Nightwish is one big swipe at their former singer Tarja Turunen.
I think Weird Al would be disqualified since parody and pastiche is his whole deal 
Explanation here.
Eminem’s “Marshall Mathers”:
An anti-Backstreet and Ricky Martin
With instincts to kill NSync, don’t get me started
These fucking brats can’t sing, and Britney’s garbage
What’s this bitch, retarded? Give me back my 16 dollars!
Later in the same song:
Vanilla Ice don’t like me
Said some shit in Vibe to spite me
Then went and dyed his hair just like me
I know the song well and don’t buy it. And even if it is a satire on the song, it isn’t a dig at Lennon, especially since it was done way later. It falls into the nasty song category that Dylan does so well.
Dylan does do song parodies - Clothesline Saga on the Basement Tapes was one of Ode to Billy Joe, and Tweeter and the Monkey Man was to Springsteen songs. But not digs at the people.
Now, in the version of “Talkin’ World War III Blues” in Don’t Look Back, he uses Donovan instead of Abraham Lincoln. That’s a dig.
And speaking of Dylan, an obvious example of a dig at him is “A Simple Desultory Phillipic” by Simon and Garfunkel. Not a good dig, but a dig.
From “All Men Are Liars” by Nick Lowe:
Nobody here knows Neil Youngs answer to Sweet home alabama? I’m not even going to tell you then
That’s what I was remembering as well. I recall an interview with her a couple of years ago (maybe with NPR’s Terry Gross?) where she finally admitted that.
Take a listen. While the songs are parodies of each original, the lyrics take a direct swipe at the respective artist.
New Radicals, “You Get What You Give”
The Supersuckers’ “The Captain” is about a failed recording session with Capt. Sensible from the Damned.
And take a look at Ronnie Van Zant’s t-shirt from the Street Survivors album cover:
https://i.ytimg.com/vi/kWROHfoONco/maxresdefault.jpg
(Tonight’s The Night shirt)
Joni Mitchell pretty clearly blasted Jackson Browne (about Daryl Hannah) in her song Not to Blame.
“From coast to coast
They said you beat the girl
You loved the most
Your charitable acts
Seemed out of place
With the beauty
With your fist marks on her face”
I Dig Rock and Roll Music by Peter, Paul and Mary.
The song mocks The Mamas & the Papas, Donovan and The Beatles.
Simon denies that it’s about Geffen on her site.
Also, her publicist denies it’s about Geffen.
In addition, Simon wasn’t acquainted with Geffen in 1971 when she wrote the song.
buddha_david beat you to it in post #14.
Damn–she really got around, huh?
Fun to play when the jukebox is spitting out a lot of country music.
There’s the classic Roxanne Wars.
English Girls Approximately by Ryan Adams is about Beth Orton.
Canibus vs LL Cool J
Second Round KO attacks Cool J due to 4,3,2,1
“Blame Canada” from SP:BL&U:
When Canada is dead and gone
there will be no more Celine Dion!
Simon & Garfunkel’s “A Simple Desultory Philippic” poked at Bob Dylan a bit:
. . .when you say “Dylan”, he thinks you’re talking about Dylan Thomas, whoever he was.
The man ain’t got no culture.
But it’s all right, ma, everybody must get stoned.
Not a well-known song, but the Old 97’s Crash On The Barrelhead is allegedly a swipe at Ryan Adams.
Mark Kozelek, from two choruses in his song “Livingstone Bramble”: