Well, if they have local software on your computer or you’re going through a SOCKS proxy or they have their own CA root inserted on your work computer and so can monitor any of your TLS communications, yeah.

But normally you’re talking to Reddit over TLS, so all that’s exposed to the network in plaintext is the domain, reddit.com – that’s exchanged in the handshake in plaintext to facilitate virtual hosts, where multiple websites live on one server. It doesn’t expose the path to the file requested to the network in plaintext, which is where the subreddit name is.

I would guess that he’s more worried about his manager walking by his desk.