Network Congestion Issue With Two Networks in One House

Need some help - I'm not completely computer illiterate, but I stopped trying to learn complicated stuff years ago. "Competent with computers" is the best way to describe my "level."

I live with my parents. They have a Comcast cable internet connection and their own LAN network. They have multiple computers connected through a router, but also wireless devices such as my XBox One. We'll call this Network A.

I have a Verizon DSL internet connection (it's the slowest broadband you can have, but I don't share it) and I have my own separate network. It has a wired desktop connected to it, as well as wifi using a laptop and multiple wireless devices. We'll call this Network B.

Whenever I try to move large-sized files from my laptop to my desktop (both on Network B), not only do I destroy my bandwidth on Network B (which I assume is to be expected and acceptable), but it also zaps all the bandwidth on Network A. In this case, I hope I'm using bandwidth correctly - both networks come essentially to a crawl or to a stop and are unable even to browse the internet. I use the network to move stuff because for some reason, my laptop does not sustain an external HDD connection, regardless if the external device is powered by AC or by USB.

Can anyone give me any insight into why when I'm moving files from my desktop to my laptop (all within Network B), Network A also becomes essentially unusable?

Network A uses the Comcast cable modem, I believe a wireless router, as well as a network extender. It has at least one desktop wired and multiple other computers and devices using it's wifi signal.

Network B uses the Verizon DSL modem/router which broadcasts wifi to my laptop and other devices and is also directly connected to a desktop PC.