In reply to Stig (but I'm really talking to the OP):
> Monsal Head is a good suggestion. That car park never seems busy.
It's rarely full but not very big, the odds are pretty good but it's not a dead cert that there will be three spaces at 10am ish. Perhaps as a plan B if there isn't space you could head along into Great Longstone.
Miller's Dale further along the trail might be a safer bet. (Guessing a bit, I've not been over there for ages.) Alternatively one of the car parks just off the A6 - Wyedale, White Lodge or even the Chee Dale layby.
White Lodge is just off the A6 at the bottom end of Monsal Dale, about a mile and a half south of Monsal Head and the viaduct. If the forecast is right about there being a chilly northerly wind blowing, it might be nicely sheltered down by the river there.
> We went to Fairholmes last week (Derwent dam) and it wasn’t particularly busy. Massive car park and further parking along the road. I’d head there.
The Ladybower area is a safe enough bet - it's very busy but there's a lot of space what with Fairholmes, the various car parks on the way there from the A57 and the parking along the A57 itself between the viaducts. The Heatherdene P&D is very unlikely to be full too. Unless the weather is dreadful it'll be busy down by the reservoir, but you don't have to walk far to get away from the crowds. Besides the more obvious walks from there round the reservoir and up Win Hill, there are a couple of steep little paths that head out of the back of Heatherdene car park directly up the hill onto Bamford Moor.
If you do go that way and find yourself coming back down Hordron Edge to emerge by Cutthroat Bridge, on no account try to walk along the road from there back to Ladybower - it's a hideous experience and you'd be taking your life in your hands. Much better to cross the road at Cutthroat Bridge and take the bridleway that skirts around the Northern side of the Ladybower Quarry before dropping back down to the road by the pub.