The cause of the blaze, which broke out at a property in Batley, is yet to be confirmed.
Wilton Park in Batley has a Green Flag award - but some residents claim it is in a poor state.
The late MP's sister says the best tribute is "not merely to repeat her words but to live by them".
A councillor says the the damage to such "important community facilities" is "really disappointing".
The Run for Jo was set up a year after the MP was killed on her way to a constituency surgery in Birstall.
Fifteen men are given prison sentences of between eight and 17 years for the attacks in Bradford.
Sandra Major and Fazila Aswat were Cox's assistants at the time and said her legacy lives on.
Spen Valley MP Kim Leadbeater reflects a decade after her sister was murdered in Birstall.
Two men are still wanted over the incident in which victim, 23, nearly lost his leg.
