The Royalists would have had more aristocrats in the cavalry while the Parliament had more yeomen but both sides had landed gentry making up their leadership and they both drew their manpower from the labouring classes of the areas they controlled.
In fact the fighting men of both sides tended not only to come from the same class but be exactly the same people.
Men regularly switched sides when whichever leader they were following lost and they got captured.
An ancestor of mine who trailed a pike under Richard Skippon was an anabaptist weaver who later became a Quaker. I suspect he was a conscript and spent the war in Northamptonshire. I cannot prove he was at Naseby but it is highly probable.
Cromwell's cavalry was largely comprised of yeomen as it was very costly to keep horses. They were not usually draught animals. Ploughing and the hauling of carts was still very much the preserve of oxen.
There were also many Hugenot volunteers from France who were highly respected as soldiers by New Model officers.
It was Cromwell who argued against an officer class - `Give me a plain russet-coated captain who knows what he loves and loves what he knows' - a clear injunction for religiously inspired leaders of men.
Cromwells army could be misleading. Although later leader of the new model army following fairfax stepping down the force that he arguably shaped the most were the troops he commanded when part of the army of the eastern association. He had been a local landowner in ely and made his name by courage on the battlefield but also by the character of his men noted for their godliness lack of drinking gambling etc and a discipline on the battlefield in contrast to royalists like prince rupert. In terms of social class there is a varied picture the london trained bands who saved parl early in the war were definitely working class. .. oliver cromwell peter gaunt oliver cromwell jc davis eng civilwar philip haythornthwaite the civil wars kenyon and ohlmeyer. Hope this helps