When we ate at the Swan, we had to pay $12 for Valet parking (was $15 but the handicapped received a $3 discount).
So last year, we went to MGM (Hollywood Studios) that day, took the boat to the Swan and returned by boat. So we didn't have to pay the Swan parking fee but did have to pay the parking fee at MGM which we would have had to pay anyway.
This year we are going to try the Downtown Disney bus (a handicapped bus though). I read that a taxi from Downtown Disney costs about $17 (not including tip)