I guess they have same Apple id on their phones...
Here is how you fix that
Go to setting
Click on messages
Click on send and receive
Remove the email, only keep the phone number
Do that on both phones.
It should solve the problems.
Are your parents using the same AppleID or iCloud?
"This is being caused by using the same Apple ID for iMessage and FaceTime on both phones. To separate your messages, go to Settings>Messages>Send & Receive and do one of the following:
Either uncheck the email address shown under "You can be reached by iMessage at" on both phones; or
If you want to continue to receive iMessages at this email address, on one of the two phones, tap the Apple ID at the top, sign out, then sign back in using a different Apple ID.
For FaceTime, on one of the phones go to Settings>FaceTime tap the ID shown, sign out, then either leave it that way or on sign back in using a diffferent ID."