That's not quite right.
"Manquer de" is used to mean "to be short of (something)", i.e. "to lack".
And "tes parents te manquent", when talking about a feeling of sadness at someone's absence, means "you miss your parents". A simple way to look at it is as the reverse of the English (though this is not entirely accurate).
The only way it would mean "your parents miss you" is if you have the opportunity to meet but the timing is off, so fail to encounter one another. This is a physical fact, not an emotional one. It's not used to mean that your parents feel a sense of lack at your current absence.
Note that in the latter instance the object would be direct, not indirect:
Four don't understand the apparent reversal that happens when translating the French into English, and the fifth is going above and beyond.
It does depend on context, but for the lone sentence given, we understand the English meaning to be that you're going to feel sad because your parents aren't with you. The French for this is "Tes parents vont te manquer."
There's more discussion here:
No, this one's not reflexive. It simply has a personal pronoun as its indirect object. The larger verb phrase to be aware of, and that we have here, is "manquer à", but if we use a personal pronoun as the indirect object, the "à" disappears and the pronoun goes before the verb.
(The indirect object pronouns in play here are "me", "te", "lui", "nous", "vous", and "leur". Whichever is used, the "à" is subsumed within it.)