No, it's much less expressive than a PDA, or even a DFA. There's a stack, but what you don't have is the ability to return to the same program state with different stack contents, or for that matter, to return to the same program state at all.
Execution is linear and there are no loops, so any given instruction in the script is executed at most once. And each instruction can only inspect a bounded number of stack or input elements. So a Bitcoin script cannot even recognize the language
x* (any number of
xs) because a script of length n will only ever be able to look at, say, 4n elements of input, and it cannot distinguish
xxxx....xy if both strings have length greater than, say, 4n+1. Even a DFA can recognize that language.