Parser combinators will eventually generate a push-down automaton. Regex would eventually generate a deterministic finite-state automaton. There's no contest, CPU-wise the DFA will parse the string faster (just have to do one transition for each char/ it's one lookup). In some cases it might take more memory for the automaton though. (also this assumes you don't go into library-specific extensions that allow backtracking, but keep within the original regex scope)
It's fine and fair to suggest that parser combinators are better for many usecases (first of all, they're clearly more powerful! they can parse any context-free language). But it's misleading to suggest they "beat" regex (they don't - not on the regex turf).
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