If you're happy with driving to Ikea park and ride, and jumping on the tram there, that would be fine. I don't drive myself but i've been down there at that time and i don't recall the park and ride car park ever being full. Meadowhall however, may fill up. As for getting back at 6pm, i'm pretty certain the tram's running at that time. If it's going to meadowhall it will stop at the Ikea stop on it's way, the same for centertainment also.
Other options include getting the X17 bus from Barnsley straight to Sheffield city centre, but i'm not sure where in Barnsley you're situated, and if you could get to the bus stop/bus station without having to take a previous bus or drive/park there.
If you drive straight in to Sheffield city centre there's plenty of parking available and by the time you've paid for a south yorkshire bus ticket or paid for the tram you might aswell just pay for parking. Parking close to Leopold Street is: Wellington St NCP, Carver Street, Eldon Street, Fitzwilliam Street (if you don't mind walking straight down Division Street to Leopold Street, not far), or Arundel Gate NCP.