Like most US rock and pop groups of the sixties, the Monkees didn’t play on their hits, they only did the singing. They were in good company: The Beach Boys, the Mama & Papas, the Ventures, the Carpenters, Paul Revere & the Raiders, the Byrds and many more relied on studio musicians.
In Hal Blaine’s biography (studio drummer from Los Angeles) there’s an amusing part about how the Monkees’ record company tried fighting the rumors that were going around in 1967, that the Monkees didn’t play their instruments on the records:
I remember one day in Studio A at RCA we were making Monkees records in a studio the size of an aircraft hangar, with all the doors locked and “closed session” signs all over. Next door, in Studio C, The Monkees sat at their instruments making music for the press. The studio bosses had set up the session to squelch the ugly rumors, and the boys gave a convincing performance. Eventually, things cooled down, and later that year I got a call from band member Michael Nesmith (he of the drawl and knit cap) asking me to contract some musicians for a few dates.
Source: Hal Blaine / Mr. Bonzai: Hal Blaine and the Wrecking Crew, Rebeats Publications. Alma, Michigan, page 86