Je n'arrive pas à croire que votre mère vous aie laissée y aller.

English Translation

I can’t believe your mom let you go.

why not “ayez” here?

English Translation

I can’t believe your mom let you go.

It isn’t ayez because the subject is mère. You is the object.

You are right that aie is wrong. It should be ait, because the subject is Mère

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What @corgwin24 said.

Besides, regarding “laissée” : the final “e” is not required:

  • Don’t think that the final “e” is required because mère is feminine
  • The question is: does “vous” stand for a male or a female?
  • Answer: we don’t know that “vous” is feminine
  • Even if it was, laissé (without the final “e”) may still be considered correct (French spelling reform 1990)

Without any context, I’d use:
Je n’arrive pas à croire que votre mère vous ait laissé y aller. (formal, if “vous” is a single person)
Je n’arrive pas à croire que ta mère t’ait laissé y aller. (more familiar)

We call it “passé du subjonctif”.

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