"based on their moral judgement" could be interpreted as, and I suspect was meant as or roughly as, "based on what they believe is moral". That doesn't imply relativism.
(I feel a bit wary about the word "subjective" because I feel like it is used in a number of different ways, and I don't want to get those different meanings mixed up. This is why I said "relativism" rather than "subjective".)

I've never been in a position of much authority over a child, so I can't speak from experience on the side of the parent, but from my experience as a former child, my expectation is that often if a child (in the sense of age, not just "offspring regardless of age") disagrees with a moral claim and instruction made by a parent (especially one endorsed by much of society), that talking about it can go fairly well, as far as the parent's goals/beliefs are concerned. The parent is more experienced in life, and often better able to compose arguments, and that combined with even a small amount of claim of authority, can I think go a long way.

Of course, there are three main types of disagreements of this kind that I can see:
1) The child believes that an action is permissible (or simply safe), and the parent believes (perhaps knows) that it is not. In this case, it seems reasonable that the child be more willing to default to the parent's perspective, because if the child is right, less is lost if they comply than is lost if the parent is right and the child does not comply.
2) The parent believes that something is morally obligatory, but the child does not believe this. Again, for the same reason, it seems like it would make sense for the child to be more likely to default to what the parent instructs.
3) The parent makes an instruction for sake of convenience which they believe is permissible, but the child believes to be impermissible. In this case it seems more reasonable for the child to resist the instruction more thoroughly. If sufficiently strongly convinced (i.e. if they have a strong enough degree of belief that they are correct) it may be appropriate for the child to refuse, even under risk of punishment, or, in extreme cases, even flee and seek refuge with others.

I guess there is another case, which seem less relevant:
4) The child believes that something is obligatory, while the parent believes that it is not obligatory, and instructs the child to not do it. I don't think this situation happens enough to be all that relevant.

However, there is also the possibility of a combination of some of these cases.
If the child believes that an action is morally forbidden, while the parent believes it is morally obligatory, then this is indeed a difficult situation.
If they cannot reach an agreement, then the only thing left to determine what happens is power, but this is not what determines who is in the right. If the child is correct, then they are in the right to refuse as best they can, and if the parent makes them do the thing, the parent is in the wrong. If the parent is correct, then they may be in the right to make the child comply, provided that they means that they do so are commensurate with the severity of the moral issue, and also not more forceful etc. than is necessary.

This all seems straightforwards to me.

When someone has practical authority over you, it is generally appropriate, insofar as is compatible with acting morally, to act in accordance with their commands-in-the-area-in-which-they-have-authority, but if their commands contravene what is morally required of you, then one ought to do what one is morally required to do (tautologically), not the commands to the contrary.

If you are in authority over someone else, you not only have an obligation not to command them to do anything immoral, but if there is something which they believe to be immoral, but which you do not believe to be immoral, and it isn't critical that they do it, out of respect for the possibility that you may be wrong, and also out of respect for their goodness (If something is morally permissible and not morally obligatory, but one incorrectly believes that it is impermissible, as long as one believes that it is impermissible, one should not do it. If you tell someone to do something not-morally-obligatory-nor-supererogatory which they think wrong, and they do it, then you have caused them to behave poorly, which is something you should not do.) it is generally better not to ask that they do it.