Unfortunately this is one of those times that the Wikipedia article is not written very well or is plain incorrect.
Sir Thomas Fairfax was at Tadcaster for the battle in December 1642 and then withdrew his forces to Selby before moving Northwest and storming Leeds during January 1643. Bradford was first attacked in December 1642 by around 1000 men under Lord Evers but their half hearted attacks were easily beaten back by local clubmen, some 50 or so musketeers, a captain borrowed from Halifax (John Hodgson) and a good 10-12 sharpshooters stuck in the cathedral tower. There's a great account of the fight in the contemporary pamphlet 'The Rider of the White Horse' which was reprinted in the Bradford Library booklet 'The Sieges of Bradford' but I no longer have a copy.
Bradford was then briefly under siege by the Earl of Newcastle in June 1643 directly after the Battle of Adwalton Moor. Sir Thomas was definitely at Bradford then and managed to break out from the town during the night and escape to Selby and then Hull.
The second fight at Bradford was in March 1644 when two parliamentary regiments (John Lamberts horse and John Brights foote) had been sent in advance of Sir Thomas Fairfax's force which was returning to Yorkshire after Nantwich. They came into Bradford via Keighley and beat up two small royalist horse regiments (George Middletons and John Bradshaws) after a short fight. There is an account in one of the books on the Civil War in Yorkshire where the defeated royalists fled to Baildon Hall but they were certainly at Hunslet on the evening after where Lambert destroyed both of them in a night attack.
The third and final battle was an attack by John Belasyse, later in the same month, leading almost the entire remaining Yorkshire army including a sizeable contingent out of Nottinghamshire under George Porter. Lambert and Bright were the two regiments defending and put up a good account until they ran out of ammunition and then decided to break out of the two. During the breakout they managed to rout Porters entire Nottinghamshire force which caused the rest of the royalist army to break contact and retreat. On seeing this, Lamberts force promptly turned back around and returned back into Bradford.
Anyway, sorry for the history lesson but Wikipedia is usually terrible for anything but the most basic information and this was one of the times it was wrong.
Having said all this, is there any way to change Bradford to Parliamentarian control at the start of the game. I think it begins as Royalist controlled which is incorrect.