If the parents have evidence that the guttering injured their child, then they may have a claim, and there will undoubtedly be a bevy of Personal Injury lawyers waiting in the wings and rubbing their hands gleefully. However, it should be covered by the Third Party Liability cover in your LL's building insurance (in the same way as if your chimney pot is blown off in a storm and wrecks your neighbour's car).
I think the only way the insurers can wriggle out of this is if they can demonstrate that the guttering hadn't been maintained and had been allowed by your neglect to fall into a dangerously unstable state.
Can we assume you do have landlord's building insurance?
'Pause you who read this, and think for a moment of the long chain of iron or gold, of thorns or flowers, that would never have bound you, but for the formation fo the first link on one memorable day'. Charles Dickens, Great Expectations