'Servicemen and women' is ok, i'd use that to avoid repetition.
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I agree that it should be expressed in parallel form; this is ethically and aesthetically proper.
Servicemen and servicewomen are both entries in the Oxford Dictionary. It honours the service of men and women in equal measure. Do not attach 'service' to men and not to women in written form.
The test is to flip the pairing: Would anyone like "servicewomen and men"? Case closed.
I agree that for parallelism, it should be service men and women because that's more grammatical and aesthetic. In addition, the logical meaning of the phrase as it stands is "male members of the military and all women, military or civilian."
But we've lost the battle, I'm afraid. If you google the US Dept. of Defense website (and others), "servicemen and women" seems to be well accepted