I actually work for a motor insurer determining liability and based on what you've said, I'd say you're not at fault. It will also depend on what information and evidence the third party (other driver) has provided though as everything is taken into account. I would ask your insurers for a breakdown of the full reasoning behind their decision along with the evidence to back up the decision.
It could be that they've said you turned in front of then to park (as vehicle damage could likely support this version of events) and without a reliable witness or cctv, it would be more plausible that you turned in front of them to park rather than they attempted to turn the wrong way into a one way system.
Claims are what can be proved rather than who was in the right, it's unfortunate but true.
In a lot of cases cctv isn't available as its only kept for 28 days and by the time the claim is passed to someone to actually investigate it, the cctv may no longer be available as we also have to have written consent from the policy holder as well as Id proof from you to even request it. Claims are likely only fully investigated when a claim is received against your policy rather than simply when you report it.
Best to request cctv yourself as soon as you're in an accident, take images of the scene, the damage and the inside of the other car. Note down any passengers and if anyone looked/complained of being injured at the time. Get witness names, emails and phone numbers too. If you can, try and get a photo of the cars still in contact with each other. Dashcam is always best but the above will help if you don't have any footage.
Insurers will 100% fully defend a claim they have enough evidence to defend. Based on your circumstances, I'd want to know what evidence the third party have and what their version of events is to make both the arbitrator and your insurers decide against you.
Hope that helps a little.