People had a variety of different reasons for 'joining-up' in the English Civil War, ranging from ideological commitments to being impressed (in the press-ganging sense) into a militia with little choice.
Although not British, Carlo Fantom (A Croatian mercenary who came to fight in the Civil War) has an incredibly poignant quote, which I think sums up perfectly many's attitude: "I care not for your Cause; I come to fight for your halfe-crown and your handsome women. My father was a Roman Catholique, and so was my grandfather. I have fought for the Christians against the Turkes, and for the Turkes against the Christians". Fantom characterises the average civil war soldier, many were involved purely for the pay, a good meal and the chance of plunder. In Thoms' book - who wrote about many exploits during the Civil War - he references how Fantom's commander had to save him from hanging twice for sexually assaulting/raping two women (one in Winchester and the other in St. Albans) "for he was an admirable horse-officer, and taught the cavalry of the army the way of fighting with horse". This was a common occurrence on battlefields of the era, and ultimately it's important we note that the chance to rape and pillage was actually a welcome one for many peasants, as it meant they could gain better fortune, certainly more than their pitiful wage could afford them.
From what I can understand, a soldier during the Siege of Taunton (1645) earned roughly £18 a week in today's money, if we look at a horde of five months pay found hidden in 2015. This would have been good enough money for a peasant at the time to justify joining not on the basis of undying support of X, but more out of the need for money. This, combined with the chance of plunder, meant you'd probably look for who pays you the most! Often, when you have not been paid your commander would have given you the right to plunder in lieu of pay - this was mainly to keep the men happy and the morale high.
You don't really have a chance to join the New Model Army, unless you are a devoted Protestant. Not only did Cromwell not accept those who weren't strong in their faith, but as Charles XII of Sweden's Carolean's show (and Cromwell recognised this), religion allowed for the men to work more cohesively and have greater faith in their ability to win as 'God's on their side'. If you did join up, however, you'd probably be a footsoldier as cavalry would have to supply their own rides.
Many soldiers refused to serve in certain armies on religious and political background, for example a Catholic would evidently refuse to fight for the Parliamentarians. However, that was if you had a choice. It was not uncommon for the devoted members of respective sides to march into towns fully-equipped and ready to force each and every man of the settlement to take up arms and come with them to the fight.
As well as this, many local landowners would often conscript their local serfs and peasantry to fight for their own personal beliefs, whatever they may be. If you're in Warwickshire in 1642, your local lord and representative in parliament was Robert Greville, 2nd Baron Brooke. He was an incredibly staunch roundhead (parliamentarian) and you may have been conscripted to fight under him at a variety of battles and small skirmishes such as Kineton and the Battle of Edgehill (1642). However, it actually depends on where exactly you're from! Prince Rupert, the right hand man of Charles I, and commander of the royalist forces pillaged many villages including Arden throughout his armies movement into Warwickshire to meet the roundhead forces. If you were in the path of this army and had not yet been conscripted into parliamentarian forces, chances are the cavaliers would have you - either by gun point or by complacency in the face of many thousands of men with weapons being rather angry at you if you don't!
The overall answer, do you have a choice? It depends - but not really, no. If you felt particularly compelled and for some reason had strong views, then you may have been able to join a faction of your choice by making your way to their nearest held areas - or waiting for them to reach your own county. Regardless, you're a peasant - and you're not going to be well-versed in the political climate of England at the time, nor will you have formed opinions far from that of your lords.
Chances are, you'd just wait to be be conscripted either forcefully or just about complacently into whichever side got their hands on you first. You'd probably be encouraged to stay and fight for the chance of pay and plunder, and hopefully you'd survive to return to your farm that's hopefully not on fire and still has a few livestock knocking around!
Sources:
Thoms "Anecdotes and Traditions"
Key Facts About the Battle of Edgehill
PS. I've missed one source that I'll add later today! I also had to save it quickly as I need to go out, but I'll check for grammar, general structure and fact check/add later also! Sorry for the inconvenience, hopefully this all helps!