I agree with Michael Rybkin. I think that "but" is used as conjuction there.
At least according to the meaning established in the Oxford Dictionary
Used to introduce a phrase or clause contrasting with what has already been mentioned.
‘he stumbled but didn't fall’
‘this is one principle, but it is not
the only one’
‘the food is cheap but delicious’
‘the problem is not
that they are cutting down trees, but that they are doing it in a
The state of being strikingly different from something else in
juxtaposition or close association.
Some parts [horse-like] are strikingly different from the other parts [eagle-like].
I have highlighted an specific example above where you can observe that the verb is used only once, the ellipsis mentioned by @Tᴚoɯɐuo
The food is cheap but delicious.
The food is cheap but (it's) delicious.
You got the same case in your example. Let's simplify
They had horse-like bodies but eagle-like heads.
They had horse-like bodies but (they had) eagle-like heads.