The Grateful Dead started performing with two drummers when Mickey Hart joined the band in September 1967. Were there any earlier rock bands with two drummers?
The only rock band I know of with two drummers was Adam and the Ants in the eighties. Significantly later than your example, so granted not much help to your question. Just thought I’d share.
Late 60’s: Mothers Of Invention sometimes (often?) had two drummers on stage.
Early 70’s: Allman Brothers Band, two drummers.
Doobie Brothers had two drummers, but the Mothers predate them, and I’m pretty sure the Allman Brothers do as well.
There are TV clips of The Strangeloves (“I Want Candy”) from 1965 that show them using two drummers - one full kit and then a couple of extra drums. Not sure if that counts or if you mean groups with two full kits. Also not sure if this particular group used this presentation just for TV.
I would have said almost the same thing, but about 38 Special (though I guess they started in the seventies).
Also, going outside rock a bit, James Brown used two drummers in the 1960s. Not sure what year he started doing that.
I have a Dick Dale album from the 90s where he has two drummers, but I don’t know if he had two drummers when he played in the 60s.
Genesis toured with a second drummer after Peter Gabriel left – initially with Bill Bruford and then Chester Thompson for many years afterwards. But that wasn’t until '76 or so.
Since it looks like this thread has already ever-so-predictably entered the “Name Any Band From Any Time Period That Ever Had Two Drummers” phase, after their initial formation in 1969 with drummer Richie Hayward (RIP, Richie) Little Feat added percussionist extraordinaire Sam Clayton to the mix in 1972, where together they would combine to drive the band to stratospheric highs at countless live shows in the following years.
Little Feat’s live album “Waiting For Columbus” stands as testament of their amazing chemistry as a showband.
Since we’ve already gone a bit off-track…
Santana used both a percussionist and a drummer as early as 1966. I’d consider a percussionist to be a drummer.
So did Crabby Appleton, though that was in 1969.
Clear Light, 1967. Their drummers were Dallas Taylor and Michael Ney.
“To fully appreciate the spectacular sound of double drumming in Clear Light, play this record at high volume.”
Well, it’s either that, or a thread with one “Nope” reply. Which would you prefer?
I used to have their first (only?) album. Didn’t care for it much. I was only interested because their bassist (Doug Lubahn) played for The Doors on the Strange Days album.
Eagles used two drummers on stage, but I don’t know when they started doing that.
The Who, of course.
I mean, I’m pretty sure Keith Moon counted as three drummers in and of himself. 
And of course, while few (if any) Latin music acts had two trap-kit drummers, probably the majority had more than two people playing percussion.
There were two drummers on the Gloria session. Did they both play? Can you hear two of them? You have to decide for yourself.
"There remains some dispute about whether Millings and McCauley were miked up, but Alan Henderson contends that Them constituted the first rock group to use two drummers on a recording.[5] "
April Wine
An earlier British artiste with two drummers was Glam Pop Rocker Gary Glitter. His twin drummer backing band went on to separate success as The Glitter Band. That was the early 1970s.
Since Glitter was revealed as a child sex offender he tends not to be mentioned these days but he was a major artist at the time and was surprisingly influential.
TCMF-2L