Disclaimer: I have not used MQTT bridging but I have researched it for connecting two Home Assistant systems (located far apart).
If you have a VPN connecting your network to your parents network, then you may be able to bridge the two brokers directly. I’m considering using Zerotier for this purpose because it seems like the easiest to implement.
Without a VPN, you need a ‘middleman’ MQTT Broker in the cloud. Two bridges are created:
Your broker <--bridge--> Cloud broker <--bridge--> Parents' broker
The ‘bridge’ is just software configuration. You instruct your broker to share topics with another broker.
There are free cloud-based MQTT services available but the ‘free tier’ usually does not offer bridging. For example, CloudMQTT’s free ‘Cute Cat’ plan doesn’t allow bridging but it is available in their cheapest paid plan. adafruit.io offers a free cloud-based MQTT broker but I can’t find anything in their documentation that discusses bridging.
Be sure to use SSL to secure the communications between local and cloud-based brokers.
The following threads discuss connecting a local MQTT broker to a cloud-based broker:
MQTT Bridge
SNIPS MQTT Bridging
This thread discusses bridging to Adafruit.io’s MQTT broker as a means of providing free SMS from IFTTT. Bridging is discussed about halfway through the first post.