I was reading an alternate history story partially about the Russian space program. It makes reference to them using tubes in 1962 (1962 being before the history diverged in the novel's universe)
In 1962 both tubes and transistors were in use. The transistors of that time were still quiet limited in sensitivity, power and frequency.
Even today an application that needs high power and high frequency, like a broadcast station, will use vacuum tubes in the final stages.
If you want a more thorough answer you might want to rephrase your question and post in /r/AskScience
Have a look at this link. One of the posters built an Apollo computer in his basement: All integrated circuits.
Same goes for the Argon computers built for the Soviet space programme - computers that were in production well into the 2000s.
Like the above user mentioned vacuum tubes and transistors were in use. In fact the US military (both army and Air Force) used vacuum tubes up until the 1970s as there was quite a plethora of equipment that was either maintained via the requirement of tubes or were built to use tubes because of higher power requirements.
As to the specific equipment that required tubes I do not know, but the US Army still printed Hickock TV-7 tube tester manuals up until 1979 and the Air Force used TV-7s up until the mid-80s with supplementary manuals being printed until 1986 it seems. With this in mind it would not be a stretch of the imagination to see a combination of vacuum tubes and transistors being used in 1962.
Edit: I'll further add that most if not all vacuum tube testers required larger vacuum tubes themselves to function, so it would seem that vacuum tubes were used in US aviation and the Army up until the 80s. As for NASA that may not be the case, but your question pertained to 1962, which I can answer with a very definite yes that Vacuum tubes were still commonly used then.
Sources:
1.Organizational, Direct Support and General Support Maintenance Repair Parts and Special Tools Lists for Test Sets, Electron Tube TV-7/U, TV-7A/U, TV-7B/U, and TV-7D/U Department of the Army, TM 11-6625-274-24P, November 1979