The general rule of law is that a gift can't be taken back, absent circumstances and exceptions that don't apply in this case (e.g. gifts by insolvent people to defraud their debtors, transfers of property made by mistake, gifts made under undue influence by someone with impaired mental capacity).
Whether a parent buying something for the enjoyment of a child in their household is actually a gift that transfers title, or does not divest the parent of ownership since the parent is only letting the child use property that continues to belong to the parent, would be a fact intensive inquiry that a court would have to make if presented with the question.
In general, modern societies aren't very careful about documenting who is the legal owner of most kinds of tangible personal property, which gives rise to the aphorism:
Possession is nine-tenths of the law.
If you take it with you when you move out, your ownership of it will soon be beyond question for all practical purposes. If your parents try to prevent you from doing so, there are few practical and authoritative ways to resolve the dispute in a cost-effective legally definitive way.