No. There were still knights then (as there are today) but the title was largely ceremonial, given out to nobility or as a sign of favor. Knighted people did indeed fight, but it certainly wasn't an obligation that came with the title.

Take Sir Edmund Verney who did fight in the war, but was knighted back in 1611 by James I as a sign of favor, not after serving as a page or a squire. Or Sir George Lisle, who was already a veteran soldier before being knighted by Charles I in 1645. There were plenty of other sir’s running around the battlefield as well, but many of these were Baronets – lesser nobles who were referred to as “Sir.” Sir Arthur Haselrig is a notable example of the latter.

You won’t find much resembling a medieval knight in the military sense, either. The closest thing you're liable to find on an English Civil War battlefield would be a cuirassier in full regalia. A cuirassier, being a heavily armored man on horseback, might at first glance look quite similar to medieval knights. Cuirassiers wore ¾ armor, meaning it covered their entire upper body along with a slatted covering for their thighs. But these were not truly knights, especially not in the traditional sense you mention in your question. Most were wealthy nobility – the only sort of people who could afford the exorbitant cost of the armor they wore.

Even these were an extreme rarity, due mainly to the exorbitant cost of their musket-proof armor. Cuirassiers persisted on the continent for more than a century after the English Civil War, but they already had one foot out the door in England when war broke out. Only two units of true cuirassiers were fielded in the entire war, the London Lobsters on the side of the Parliamentarians, and the personal life guard of the Earl of Essex, who fought for the royalists. Plenty of other units were designated as cuirassiers, but most were so lightly armored that they really didn’t qualify. The full cuirassier armor, while effective, was so dreadfully heavy and cumbersome that it was sometimes even discarded by those who could afford it. Philip Haythornthwaite mentions a few anecdotes involving cuirassiers during the civil war in his book, The English Civil War: An Illustrated Military History.

The aforementioned Edmund Verney was so fed up with the weight and restrictiveness of the armor he once remarked, “it will kill a man to serve in a whole Curass. I am resolved to use nothing but back, brest (sic) and gauntlet.” He was later killed at Edgehill wearing no armor at all. On the opposite side of the spectrum, Sir Arthur Haselrig (the Baronet) was assaulted by an overwhelming flurry of musket fire and melee attacks after his horse was shot out from under him, but survived in his armor long enough to be rescued. Charles I later made a joke about it, saying that if Haselrig had provisions, he could have survived a seven-year siege in that armor. But not even its effectiveness was enough to overcome the cost and cumbersome nature of cuirassiers, which faded out of use as the civil war drew to a close.

In addition to Philip Haythornthwaite’s book, Stuart Reid’s All the King’s Armies: A Military History of the English Civil War is another great source for civil war era military history.