(To) eat is an infinitive, eating is a present participle. Spanish infinitives can be translated to mean either in English, because you are translating a concept, not specific words. "Parar de [infinitve]" simply means to stop (doing) a certain verb, which in this case is (to) eat.
I posted the above before I saw someone's claim that this was so. I will wait until they provide a link as proof and I will do more research.
But my claim is correct: there are many examples of infinitives used as gerunds with or without de:
Nadar es muy divertido (swimming is very entertaining). Me gusta nadar (I like swimming). Voy a nadar (I'm going swimming). Etc..
In that last one, ir always takes "a".
I do not believe this is true.
The preceding nouns determine what prepositions/participles follow. Some require de, a, en. You are the first/only statement I have ever seen make this claim. And there are many counter examples of verbs that need NO preposition/participles at all.
Please provide a link. I am happy to be corrected.
That's not correct. de in this sentence belongs to paro. As @elissaf1 says, whether or not there is a preposition between two verbs depends on the first verb, not the second.
Certain verbs, including parar, must use de when followed by an infinitive.
A list of such verbs can be found here: http://www.elearnspanishlanguage.com/grammar/verb/verbswithprep-de.html