What was more likely was that maybe a troop within a regular cavalry regiment was equipped as cuirassier, or maybe a detached troop served in it's own right (Kings Lifeguard comes to mind here).. its entirely possible that officers, professionals, and those who could afford it might have worn full cuirass within a regular cavalry - but there were very few full regiments - in fact the only one I can think of is Haselrige's Lobsters [clicky]....
If I want to deploy cuirassiers then (and who wouldn't? it's like having an ACW project without zouaves, or a Napoleonic project without the Old Guard

So as we have read previously [clicky] Sir William Balfour had detached his troop of cuirassiers to serve in Essex's cavalry reserve, also in that ad hoc unit were Essex's own cuirassier troop under the command of Sir Philip Stapleton, Reid also believes that a third troop of cuirassiers, those of Essex's general of horse, Bedford, were also present... Reid notes that all three troops were large (he calls them "oversized") so that is enough for me to represent a regimental strength cuirassier regiment to serve Parliament as part of the forces present at Edgehill!
So what of their performance at the battle?? Very good I would say... As we have read previously, the Parliamentary horse on the flanks of Essex’s army had been driven off comprehensively, but the cuirassiers in the reserve, remained on the battlefield, in the second line, behind Meldrum’s infantry brigade.
When the Royalist infantry advanced, they charged, and while Sir Nicolas Byron’s brigade held, Feilding’s brigade of Royalist foot was handled roughly (Feilding and two of his colonels, Stradling and Lunsford, were captured, though Fielding was later rescued on the battlefield). Carrying on through the infantry, Balfour’s troopers then overran the Royalist heavy guns, but possessing no nails, they were unable to spike them so instead cut the traces on the guns (stopping them from being moved) and fell back to their position in the second line.
Drawing on the initiative that the cuirassiers had given him, Essex launched another attack on Byron’s Brigade, this time with Robartes’ and Constable’s regiments of foot, supported by the cuirassiers and the foot regiments of the Lord General and Brooke. The attack was successful, and drove it back, breaking up its ranks.
Good result all round..
Peter Pig 15mm figures, painted April 2020
Sources:
https://www.britishbattles.com/english-civil-war/the-battle-of-edgehill/