Translation:It's fun to have parties.
I think the distinction between the infinitive and the gerund when translating from Japanese isn't that clear cut, and the way you've phrased your comment is a little misleading.
For instance, "It's fun to party" and "Partying is fun" are both acceptable translations of「パーティーをするのは楽しいです」and arguably have the identical meaning in English. (A subtle difference in nuance might be a tendency to use the gerund when one is doing said action, but this is by no means a hard and fast rule.) The の has nothing to do with whether to translate using the gerund or the infinitive, but rather the way Japanese "noun-ify" their verbs. English has gerunds and infinitive, while Japanese has するの and すること, neither of which are one to one translations of the other.
Gerund in English is specifically using the "-ing" form of a verb as a noun. For example, "I like studying Japanese." In this case, "studying" is referring to "the act" of studying, which is a "thing" or a noun. In cases like this, you would use するの or すること in the Japanese sentence.
I would usually refer to the "て+いる" structure as "be verb-ing" since it typically corresponds to the present progressive, or present continuative, tense. For example, "I am studying Japanese now." In this case, the "am" (or "be" helping verb) + the "-ing" form indicates that the action is currently in progress and/or continuing in the present.
I think they are more or less the same, though using の is seen as less formal, because it is like an abbreviation.
The reason your friend felt uncomfortable is because there isn't really a hard and fast rule for deciding which one to use; rather a few different rules depending on the type of verb you're applying to the normalized verb. In some cases, you have to use one or the other, but in most other cases, both can be used. It's hard to give a concise explanation of the different rules, and in the end it's just something you have to memorize as you learn new verbs.
Kind of a bad question... Because there's no clear direct translation, there needs to be quite a bit of possible correct answers. I said "going to parties is fun" which could be considered incorrect because there's no direct implication of going to one in the sentence... But one could easily infer that meaning from the Japanese.