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Recently I accidentally reversed one of my cars (with Full Cover insurance) into my other car (with 3rd Party insurance) in my driveway. The damage to the Full Cover car is minimal, i.e. barely worth making a claim for, but the 3rd Party car sustained some fairly expensive damage. My insurer tells me I can't make a claim for damage to the 3rd Party-covered car because I can't be both the 1st and 3rd Party in an insurance claim, as there is no one to recover costs from. My understanding was that the Full Cover policy would cover damage done to other cars and property, regardless of whose car/property it was. Do I have any recourse here or am I totally out of luck?

In this case, legally you cannot claim (you cannot take yourself to court) and your insurance company has advised you accordingly. You are out of luck.