'we will go to the the park until you having finished your homework. ' she told her son. 點解會用 having finished?

This is past participle, Or absolute phrase:-
On rearranging,
Having finished your homework------past participle clause,adjective clause qualifying the main subject "we".
we will go to the park-------main clause.
A participle is a word formed from a verb and used as an adjective or to form tenses; ie:-present participle, past participle.(=having finished...)
A simplified version would be:the past perfect tense
The perfect tense is the tense of a verb formed in English by the auxiliary verb "have" and past participle of the main verb(=finish) to refer to conditions existing, or actions completed(=finished)
(a)before the present (When you've finished HW, we shall go to the park.)(have finished)
(b)before a point in the past (After you had finished your HW, we would go to the park.)(had finished)
(c)before a point in the future(We shall have left for the park by time you would have finished HW.)(would have finished)

The sentence is wrong in grammar.

Consider rewriting it as:
'We will go to the the park after you finish your homework, ' she told her son.