As Alex said, the United Kingdom is already a unitary state.
Now, that being said, there are definitely many ways to alter England's development. Parliament's ascendancy isn't predetermined—though it began to find a voice by the end of Elizabeth's reign, the crown still had a fair amount of powers. Plenty of the other monarchies in Europe had to deal with similar assemblies—such as the Spanish Cortes, which had control over taxation, and the Estates General, which served the French monarchy as an advisory body and gave fiscal advice. The Holy Roman Empire had the Reichstag, Poland had it's notorious Sejm. So the English Parliament isn't really a totally abstract body.
The Stuarts were not able to deal with Parliament as effectively as the Tudors, and a different outcome in the civil war would definitely throw a wrench into Parliament's rise with the crown. I don't think Parliament is going to go away totally—but with a Royalist victory you'd probably see the Star Chamber remain as an important organ of royal power. Charles I used it as a Parliamentary substitute, and while it was notorious by the time of it's abolition in 1640 for it's arbitrary acts of power, I think a stronger monarchy is going to need such an organ of government. If the Royalists do win, it's likely that Parliament is going to be neutered—the Commons especially, though it'd no doubt effect the House of Lords, too. They'd certainly lose any control over fiscal policy and taxation, though given how much the crown needed the gentry to collect taxes in the first place in this period, I'm not sure what they might do. Maybe they take a page out of the French playbook and create some form of Intendant to control royal policy in the counties?