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Isn't it cheaper to return the hirecar for the week you are cruising? Barcelona is like every big city where longterm parking is expensive. Have you looked at the airport costs.
We are flying from Heathrow next month and booked the Renaissance hotel (right on the perimeter of the airport) double room and two weeks parking for 83 pounds through simply-hotels.co.uk....so they may be worth a try...I don't remember if they had family rooms though.
Try www.holidayextras.co.uk they usually have good deals for hotels with parking. I think that the Novotel and the Crowne Plaza hotels situated on the Heathrow perimeter offer quad rooms and parking. The shuttle bus to the terminals picks up at both hotels.
I think you mean parking at Heathrow and not at Barcelona if this is correct I would try Crown Plaza or Holiday Inn mainly because they have rooms to sleep 4 and the price is per room per night and not per person.