I think the company that you're looking for is Framon motors. They have been advertising in the free papers about their airport parking service. I thought I'd brought the advert home with me but I'm unable to find it just now.
From memory I think it's called autoproof parking or something like that, it's a full page advert and it says "800 customers can't be wrong". I can't remember which one of the papers it was in, maybe more than one. If you search all of them either on-line or whilst you're out there i'm sure you'll find it. The cost is 250e a year, you leave the car at the airport, they pick it up and return it to which ever airport you use next time.
Framon motors is in Bigastro on the way to Orihuela, you could always drive there and have a look at the facilities.
The papers you're looking for are Round Town News, Coastrider, Euro Weekly all of which can be read on-line.