Can my work computer sniff the websites that my parents personal computer go through? I know this would be very unethical for the company to do so but I never realized this is POTENTIALLY possible when my parents brought it up when they saw me working from home.
If your employer has set up your work computer to create a security hole in the network you are connected to (like installing a remote access trojan or other reverse shell backdoor) then your work computer could allow your employer to access the work computer as if they are sitting at it. Once they have access, they can attempt to traverse the network that computer is on by exploiting vulnerabilities in services and applications running on other networked hosts. If your employer manages to log into your home router, they could download logs of all host requests on your network (what sites every host has requested). Also, once someone is backdoored into a system, they may be able to run wireshark or a tcpdump to sniff traffic to/from that host, depending on their access level. So, it's possible, but is a multi-step process.
If you suspect something like that may be happening on your work computer, open a terminal or command prompt on your work computer and try a netstat command with different flags (like "nestat -an") to see what your work computer may be communicating with and on what ports. You can also run Wireshark to see a stream of network-based communication that your work computer is doing.