All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Your statement to your wife: “If you meet your mother you will have divorce 3 times at once” is considered a promise of divorce, so this does not lead to divorce unless you intended divorce by saying so. Since you did not intend divorcing her by saying so, then there is no problem (divorce did not take place).
It should be mentioned that what should be taken into account in the matter of divorce is the statement or intention of the husband and not the statement or intention of his mother.
On the other hand, we do not know the reason why your mother orders you to prevent your wife from meeting her mother, or that you prevent your wife from meeting her mother, while a group of scholars are of the view that a husband has no right in preventing his wife from visiting her parents unless there is harm and disadvantage in her going to them.
Finally, it should be mentioned that when the wife is happy and not worried about her parents, this leads to the happiness of her husband and his children. Besides, the husband should help his wife perform acts of good deeds among which is her being kind and dutiful to her parents.
Allaah Knows best.