This was a time when computer networks were a novelty to me. The unique aspect of this setup was that the required cable could be made at home. The ability to access and browse the filesystem of another Amiga through PARnet was a special experience, especially given the technological context of that time.
That is very cool! Not Amiga but Linux: back when I was still on dialup my 486 desktop PC had a PLIP (Parallel Line IP) connection to my crappy laptop and so my brother and I could both be using the Internet at the same time. The laptop was so old and underpowered that it'd just be running a X11 server and the 486 desktop would be running the graphical session (including the browser) shown on the laptop.
IIRC the cable was just a regular parallel cable (the same kind we'd be using back then for printers).
It also felt like magic back then: one computer running two browsers, with one displayed locally and the other remotely over a parallel cable... All the while monopolizing mom's landline for the dial-up connection!
Sounds more like something called a "Laplink cable" back then (2 25pin connectors with RX/TX crossed), named after the application made for exactly that purpose you used it for. Parallel-port Printers used a different connector on the printer-side...