In terms of grammar, either form is perfectly acceptable. The preferred usage will certainly vary by region, but in my personal experience, the gerund is definitely the more common form, whereas the infinitive can often sound dated and awkward. However, I have certainly heard the infinitive used well, especially in more formal contexts.
The active voice is speech that places the subject first, followed by a verb, and a place/object. Therefore the correct form is subject -- verb -- object.
The passive voice is speech that places the object first, followed by a verb (tense and a gerund), and the subject. Therefore the correct form is object -- verb -- subject.
None of these sentences are in the passive voice, since I (the subject), always comes first in each of them.
Most of these can be converted from active voice to passive voice:
I saw a person repairing my car. The car was being repaired by someone.
I saw a person repair my car. The car was repaired by someone.
I saw a car being repaired. / I saw a car be repaired. A car was being repaired.
I watched a man solve the problem. The problem was solved by a man.
I watched a man solving the problem. The problem was being solved by a man.
I watched a problem being solved. / I watched a problem be solved. The problem was being solved.
Notice that none of these use I, mainly because I couldn't find a logical incorporation. I suppose you could say (and use as a guideline):
The car was being repaired by someone I was watching.
The problem was solved as I watched.
The sentences I saw a car repaired and I saw a problem solved don't make sense to me because the past participle is just by itself. I would include words to tie it together with everything else:
I saw a car that was repaired.
I saw a car that had been repaired.
I saw a problem that was solved.
I saw a problem that had been solved.
I also think that adding a comma, though strange, is acceptable:
I saw a car, repaired.
I saw a problem, solved.
Best Answer
The first option is correct.
The reason for that is the second sentence is actually two gerunds nested inside of each other. Let me show you what I mean.
The bolded text is the direct object of the sentence. "Watching them play" is a gerund phrase. "In the park" is just a prepositional phrase, and like most, it can be removed from the sentence without taking away anything from the grammatical structure or correctness. "I love watching them play." is also a valid sentence and a complete thought.
"Playing in the park" (or I suppose more technically just "playing") is a gerund phrase on its own, apart from "watching them". So what you are effectively saying is "I love watching them [noun]." As you probably know, nouns cannot be used as verbs.
This is a correct sentence because "playing" is a participle used to describe "children," but it is a little awkward to say so you will not likely hear or see it in normal conversation. A more common version would be "I love watching children at play." I just wanted to play devil's advocate and show you what it would take to make a valid sentence with both "watching" and "playing" in the sentence, while preserving the meaning.