Am I correct in thinking that, with Mouldridge being on CROW land, there is no need to call the farmer first or is that still done as a courtesy?
Also, I presume that visitors park on the wide grass verge across the road from the gate (right hand, small gate) taking care not to block access?

You are correct on both counts - park on the verge/hardstanding  opposite, there's plenty of room to do this without blocking the track, then just cross the road and wander down the valley to the entrance - no need to call at the farm.

In addition to the above, you can get up to date access information via the DCA. See : https://peakdistrictcaving.info/home/the-caves/bradford/mouldridge-mine

Thank you both. I did look at the information on the DCA site but I thought it was out of date, as it gives the farmers number to call and says there is no right of way down the dale. I presume that has changed with the CROW act?

You are correct that the DCA Registry is out of date, I'll update that in few minutes. Thanks for flagging it up.
Access to the dale on foot is fine as it is on CRoW land. Mine access is not covered by CRoW but there is no need to call the farmer in advance as there are no restrictions to entering the mine.
EDIT - updated now - https://peakdistrictcaving.info/home/the-caves/bradford/mouldridge-mine

Nice little trip that one