Even for a girl, we say:

Quand j’étais enfant, …

enfante IMHO is not the feminine form of enfant. cf CNRTL.

For this one:

Quand nous étions enfant, nous aimions …

I would say that you have to use enfants (plural). I don’t really see a reason that would allow you to escape the agreement – unless maybe if you considered “être enfant” to be a fixed expression. But I don’t know if you would have a very strong case. However, ngrams shows that both are used, with a very strong preference for enfants (plural). The references given by ngrams don’t seem to point to the singular being only a mistake. It seems to be a true usage.

Same analysis for parent, except that parente is this time the feminine form of parent (CNRTL).

This same question applies to a wide range of questions that appear to have a set gender but are clearly used for people of different genders: enfant, parent, but also professeur as it once was (professeure has existed only “since the end of the 20th century“). In fact, the word personne itself has seemingly the opposite issue for men — your average Joe is une personne. 🙂

The grammatical term for this kind of word is “epicene” and

… can be used in two distinct situations:

• [when] the same word can refer to both masculine and feminine antecedents …

• [when] a noun or adjective has identical masculine and feminine forms
  …

(Wikipedia)

The first case would be like personne. No matter the gender of the person meant, the determiners, adjectives, etc. will always agree with personne as feminine. (At least AFAIK.)

Dictionaries often refer to this as simply “nm” or “nf”, as the case may be.

The second case would be where the form is identical but the grammatical gender changes, and hence surfaces in words that agree. Enfant is actually the very example Wikipedia gives. It is possible to say un enfant or une enfant, and likewise un petit enfant vs. une petite enfant.

Dictionaries often refer to this as “nmf”.

The third option is that there are two corresponding forms of the same word, of which one is masculine and the other is feminine. French is full of this kind of thing too. An example would be chat and chatte — or even phonologically dissimilar variants like acteur and actrice.

Dictionaries often refer to this as “nm/nf” and/or list both variants (e.g. “ours, e”).

If you’re unsure and your dictionary doesn’t list it clearly, then as Frank has done, you can distinguish between these cases using Ngrams. For example, parent is listed in Collins as “parent, e”. If you were trying to verify this independently, you can show that une parente is well attested but une parent is not:

This rules out the second case (“nmf”).

However, it’s a little harder to distinguish between invariant words and words with two forms. Une parent is out, but how do you know that you wouldn’t say “C’est un parent” where ce refers to a woman? It certainly helps to know that une parente exists; if it didn’t, we could assume parent was an epicene word. But between une parente and an invariable un parent, I would say it’s hard to know what significance the 0.00005% usage of une parente has in the bigger picture without a way to disentangle the antecedent in phrases like “C’est un parent”.

For the plural, the closest parallel term is probably indénombrable. (English count/mass noun is nearly the same thing, but doesn’t, to my knowledge, include an equivalent for nouns like des pâtes that are plural but still cannot be individually counted as e.g. deux pâtes.)

Luckily, nouns for people and animals in French tend to be unambiguously dénombrable.* So if your subject is plural, as nous is, then enfants is the one you want.

* Except, of course, for “le monde” and “les gens”!