There are a couple at Diggle - one looks pretty new and is on the right as you take the turning off Huddersfield road, I know that they do part livery. I'm not sure of their number though so you'd probably have to just call in. There is also another yard in Diggle (again I don't know the name sorry) but I'm not sure if they do part livery.
There is High Moor stables at Scouthead - I know that they do full but not sure about part.
All of those have decent facilities and good hacking.
There is also the centre at Grains Bar near the Black Ladd - I know that they do part livery but I'm not sure about hacking routes from there unless you can get straight onto Crompton Moor.
The first one Madlady is talking about is Saddleworth livery stables, not sure which other one in Diggle mentioned is, I can think of Rhodes up at the top of Digglea near the school, but only does DIY nowadays, and Saddleworth stables (also a riding school) near the rifle range. Highmoor is Liz Longden's, they do part livery. The one near the Black Ladd only does DIY and is Liz's daughter, Emma's yard.
For full/part livery I'd go for Vicky and Graham Tibbot's yard on Lark Hill above Delph/Diggle. Nice yard, great care, superb hacking. Very experienced BHSII on site. Otherwise Debbie Syke's Hole House Farm livery near the Royal George in Mossley/Greenfield (quite near the public arena at Friezeland where RC shows are held).
I have numbers for some of those yards if you are interested.