I have:
Royalist
Earl of Derby
Tyldesley
Sir Gilbert Hoghton
Lord Molyneux
600 Foot brought from York by Tydesley being 50:50 pike to shot (not certain but I'm tying it in with the troops he brought a couple of weeks earlier) plus clubmen (I'm estimating 3,000 of these). 11 troops of Horse so lets call that 500 -550 which may include dragoons as muskets were captured at Sabden Brook. Plus 1 piece f artillery. I don't see unwilling clubmen being allowed to form part of the scouting force under Tyldesley so I don't think it would be from them although the Discourse talks of capturing foot its unclear if that was up above Read or later at Whalley itself.
Parliament is less clear
Colonels Shuttleworth and Starkie (but they may be back in reserve). The true relation mentions Captain Ashheton (not Colonel so not Assheton of Middleton, possibly Assheton of Downham?) The author of the True relation claims to have taken a leading part. his initials were EF but I haven't tied him into anyone specific. plus a Lt Marsden who came in with other troops for the pursuit down to Whalley.
The foot were assembled at Padiham so probably at Gawthorpe Hall and appear to have been around 400 or so plus around 250 horse. The true relation says two troops at Dunkenhalgh Hall with a further 150 arriving at Read Bank prior to the ambush. But it does state that as being "at first" which suggest further troops may have come in as the day progressed. The Horse may not have been of good quality as Seaton's horse a couple of weeks earlier were described as "who durst not face the enemie".
The Discourse talks of a piece of artillery firing two or three shots upon the Tower, which suggests that Parliament were firing at the tower of the derelict Abbey at Whalley. It isn't clear and none of my other sources mention it.
Just to confuse matters Lancashires Vale of Achor talks of the Earl of Derby with 2,000 men crossing the Ribble to get to Whalley so there is no uniformity on numbers apart from parliament being badly outnumbered overall.