IMO, Dr. Strangelove makes fun of everybody.
"Now then, Dmitri, you know how we've always talked about the possibility of something going wrong with the bomb. ... The bomb , Dmitri. ... The hydrogen bomb! ... Well now, what happened is ... ah ... one of our base commanders had a sort of ... well, he went a little funny in the head ... you know ... just a little ... funny. And, ah ... he went and did a silly thing. ... Well, I'll tell you what he did. He ordered his planes ... to attack your country... "
"What kind of suit do you call that, fella?"
Captain Mandrake: "What do you mean 'suit'? This happens to be an RAF uniform, sir."
Russian ambassador about the USSR Fearless Leader: "Be careful, Mr. President. I think he's drunk."
"Survival kit contents check. In them you'll find: one 45 caliber automatic, two boxes of ammunition, four days concentrated emergency rations, one drug issue containing antibiotics, morphine, vitamin pills, pep pills, sleeping pills, tranquilizer pills, one miniature combination Russian phrase book and Bible, one hundred dollars in rubles, one hundred in gold, nine packs of chewing gum, one issue of prophylactics, three lipsticks, three pairs of nylon stockings. Shoot! A fella could have a pretty good weekend in Vegas with all that stuff."
Including the ineffective liberal American President Merkin whom I believe was based on Adlai Stevenson.
I ignored the anti-nuke message in the movie because I really liked the love story between Dwight (Gregory Peck) and Moira (Ava Gardner). I loved the scene when she is walking on the dock by the carrier.
It does poke fun at the military -- but only that part of the military that is ruled by politics. The actual soldiers are intelligent, competent, and completly loyal to their duty.
(well, most of them, anyway)