My MacBook Air is receiving messages from my parents' phones. I want it to receive messages from only my phone though!
In Apple world if you are using same Apple ID, your messages and information are shared across devices.  That is the reason why you are seeing the messages sent by your family members on your macbook.  

I would advise you to create separate Apple ID for every individual or disable iMessage from their personal devices. So that the device that was disabled for iMessage would not sync messages across devices.

Hope this helps!