All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
This matter concerns your parents, and they have the right to consult whomever they wish and leave whomever they wish out of it. What your wife mentioned, that it is not permissible for them to talk to you in private and tell you some secrets without informing her, is not true. She has no right, and no one else has the right, to know the secrets of your parents.
The evidence that they have the right to these secrets is what was reported about the emphasis of the prohibition of disclosing secrets; Jaabir ibn Abdullah, may Allah be pleased with him, narrated that the Prophet said, “When a man tells something, then he looks around (i.e. departs), then it is a trust.” [Abu Dawood]
Al-Hasan Al-Basri, may Allah have mercy upon, said, “It is treason to disclose the secret of your brother.”
Besides, Al-Ghazaali, may Allah have mercy upon, said in Al-Ihyaa’, “Disclosing the secret is betrayal. It is forbidden if it involves harm, and it is wickedness if it does not involve harm.”
Your wife should be reminded of what is mentioned in these Islamic texts, and it should be clarified to her that it is better for her not to interfere in a matter that does not concern her.. Perhaps if she interferes in these matters which are exclusive to your parents, some kind of affliction would befall her. The Prophet said, “A sign of a man's good observance of Islam (his piety) is to keep away from that which does not concern him.” [At-Tirmithi]
Allah knows best.