Contrary to common misconception, and regardless of the "heritage" of some units, the constitutional foundation of the British Army has nothing to do with the Civil War – except that the "rule of the major generals" engendered a general distrust of standing armies – and everything to do with the Glorious Revolution.
The army in its current form dates entirely from the reigns of Charles II and James II. In fact, no British infantry regiment (with the exception of the Royal Scots, discussed below) was raised prior to 1661.
Even when units with Parliamentarian heritage – only Monck's Coldstream Regiment and, depending how you look at it, the Blues and Royals – were taken into the King's service, they mustered out and were then engaged as a "new" unit. The Royal Scots – Pontius Pilot's Bodyguard – which was raised in 1633 by Charles I spent the Civil War and Interregnum in French service, and only returning to England in 1661 to fill the game between the disbandment of the New Model and the creation of a new army.
When James II was king he sought to expand, mainly in 1685, and keep a standing army and this was perceived as his way of imposing his unwelcome will on the country, particularly with respect to religion. The deep distrust of James caused the Glorious Revolution and, on the accession of William & Mary, Parliament insisted on further limiting the power of the Crown in the Bill of Rights which states, inter alia : "That the raising or keeping a standing Army within the Kingdome in time of Peace unless it be with Consent of Parlyament is against Law."
William further expanded the army in 1689 to fight James and several regiments were raised subject to the Bill of Rights provisions at this time including my own, the 22nd (Cheshire) Regiment of Foot, raised on the Roodee at Chester by the Duke of Norfolk.
The Bill of Rights is still good law and the consent of Parliament is granted at least every five years by an Armed Forces Act – there is, in fact, a new one due by 2016. The effect of the provision and subsequent acts is to maintain the Army under Parliamentary control and accountability although the soldiers themselves owe their allegiance to the Sovereign.
This is actually a fundamental of English constitutional law and while it is correct to say that the Royal Navy was for Parliament, this really just demonstrates the misconception. The Royal Navy and Royal Air Force are on a completely different constitutional footing.
If you're interested and would like to learn something about the founding of the British Army, there are some very good books – one of the most fun to read is Redcoats and Courtesans: The Birth of the British Army (1660-1690) by Noel T. St. John Williams which appears to be available through Amazon for just a few pounds/dollars.
