In the good old days you could park down this little snicket (cars were narrower then of course ). https://www.google.com/maps/@53.375...4!1sc6gcBXFEuyqZQ_zPOk6WYw!2e0!7i16384!8i8192
No bother. I'll be honest, in the 2 years I've lived in Sheffield, I think I've parked in the centre twice, so I'm no expert. I previously lived walking distance to the centre, and now I live on a tram route.
I used to live on Thornsett Rd, Nether Edge which is quite posh and about 20 mins walk from bumhole lane. There's some nice streets around there.
Train is your best bet, but if you're driving there are a couple of car parks either side of the Moor...be careful because some don't accept cash anymore.