It depends. According to clinical studies, alas, pills of parsley extracts including chlorophyll don't work. I think the reason it was once an attractive idea is because theoretically, chlorophyll might stick to some aromatic molecules in your gut, by intermolecular attractions, facilitating their exit through the back passage. But just because you can think of a molecular mechanism does not mean that it works in people. If that were true, all we would have to do is think up mechanisms and we wouldn't have to do any experiments! In any case, swallowing the stuff does not seem to work.
Chewing parsley leaves, on the other hand, mechanically scrapes your teeth and helps remove potentially stinky or otherwise unsociable food bits from them. This is similar to chewing sugarless gum. This is nice if you don't have a toothbrush handy. Clean, washed parsley sprigs make a nice healthy snack, anyway.
Popping any sort of parsely or chlorophyll pill, on the other hand, does not seem to work according to human studies. And I don't think the placebo effect will help you where halitosis is concerned.