In formal prose, ambiguous referents of pronouns are to be avoided.
In informal writing and speech, ambiguous referents occur frequently. When such ambiguities arise, they cause no problem if context resolves the ambiguities.
In your first example, the context resolves the ambiguity.
In context, sensible people will interpret
They are very happy to get lucky money from them
The children are very happy to get lucky money from their parents
The parents are very happy to get lucky money from their children.
In your second example, context is not helpful in deciding whether the final "they" refers to children or parents.
There is a rule of interpretation or construction that says to choose the immediately preceding relevent referent, in this case the parents, but what that rule says may differ from what the speaker or writer intended. Moreover, a listener or reader may fail to apply what is a very technical canon of close construction if it is intended. It is best to avoid relying on the rule and to use the noun rather than the pronoun.
Their dad and mom are in Chinese clothes. The parents like different Chinese clothes<
Their dad and mom are in Chinese clothes. The children like different Chinese clothes<
These are admittedly awkward sentences, but they reduce ambiguity. There are various ways to put sentences together that are less awkward and will prevent any ambiguity.