I don't think you would be able to put your parents' home in your name for the purpose of avoiding care home fees. I believes it is called something like deprivation of assets - and, as has been said, there are checks made on this.
In my opinion, it isn't really fair to expect to make no contribution towards care home costs when there is value in an estate - and people are living much longer now so some sort of contribution really needs to be made. However, I think the current system is very unfair too. To call the OP "grabby" is, I think, nasty. Most parents want to leave something to their children/grandchildren and some children/grandchildren may really need it, especially these days with the price of rented and owned housing.
I don't know why there isn't some sort of insurance system specifically designed to resolve this issue. I know we have national insurance but the demands on this are enormous. I would be quite happy to pay an insurance that would cover care home costs in the event that care is needed. Some might say "I may not need it, so I would have spent that money unnecessarily". But you could use that argument for any sort of insurance - house, house contents, holidays, etc, etc. I would count myself as lucky that I didn't have to make such an insurance claim! It gives peace of mind, and it is there if you are unfortunate enough to need it. Something must be done about the care home fees issue. They are extremely high, and it must be awful to see the greater part of a parent's estates value rapidly disappearing.
I believe it is the case that the number of elderly people requiring admission to a care home is far less than people assume, so an insurance scheme - whether private or government - might be the answer.