As has been said, proving where your mobile phone was won't prove where you were.
Mobile phone companies do keep records of which masts have been 'pinged' by which phones but they're normally only made available to the police and security services upon production of a warrant. You can't normally access the information yourself. However it's possible that a phone company might release the information under a 'data access' request, as defined by the Data Protection Act, but they'd be entitled to charge you a tenner for it.
It would make more sense though to require whoever issued the parking ticket to provide proof (usually in the form of photographs) that your vehicle was parked in contravention of any relevant rules.
It's the driver of a vehicle, rather than the registered keeper who is (in the first instance) responsible for paying any parking penalties but (under the provisions of the Protection of Freedoms Act 2012) the charge can be transferred to the registered keeper if he/she fails to provide details of who was driving the vehicle.