The only real restrictions are on the upper decks of the piers, you are only supposed to fish off the bottom deck, but as you are after flatties, and in the summer the piers are full of "macky" bashers you want to be fishing further into the harbour.
I don't know if you are familiar with Whitby, but the one side of the river (west)is all bingo and candy floss, while the other (east) is much quiter. This is the side most of the fishind is done from.
From the sea into the harbour on the east side the first mark is Tatehill beach just round the corner from the pier. It's actually quite steep so you don't need a big cast. The club I used to be in had it's biggest coalfish come from that beach, on a twenty yard lob. The trouble is in the summer during the day the beach gets full of people, so evening would be best.
On the up stream side of the beach there's a stone jetty that you could fish off, at any stste of the tide. Next to that and behind the rowing club is the lifeboat jetty, again you could fish at any state of tide. In between the two jetties is another small beach, which is covered at high water but again can fish well on the flood.
Further upstream you've got the swing bridge the only bridge in Whitby. If you follow the main road round to the right still on the east side you'll find a tackle shop. almost opposite that there's a slip way which leads down to a muddy beach at low water, this mark can fish quite well again form half way to high tide.
As far as the west side goes it gererally too busy in the summer and best left alone, but just on the upstream side of the swing bridge on the west side is "kiddies corner" you'll see loads of kids with hand lines catching crabs, however it can fish surprisingly well in the summer for flatties eels and billet (coalfish) but you'd have a lot to put up with.
While still on the west side there is one of my favorite marks which is further upstream from the bridge. There are some big (and expensive) car parks near the railway station, behind a co=op store. By the river there's some public toilets to the left of them is a grassy bank that slopes down to the river, it's on a bit of a corner behind the custom dock, I've had some good flatties there too. Again the incoming tide is best, because the water is starting to get quite fresh that far up at low tide.
There are more marks even further up stream even under the fly over but I havn't fished them.
As I said before fresh peeler is by far the best bait, fished on a simple size 2, two hook paternoster. You'll usually get away with a 2oz bomb. A light beach rod or a carp rod will do, and keep the bait twitching every few minutes.
I know I've gone on a bit and I hope it's some use. I,ve had some very nice days there. The biggest problem youll have is parking. Finding a space in town is a nightmare, and very expensive. On the other hand you can park up top and walk down, so travel light.