Originally Posted by
rustykettel
The Keikyu limo bus runs between HND and Osanbashi Pier according to their timetable
http://hnd-bus.com/airport/h-yamashita/ I haven't ridden this one but the orange Limo bus has a limit of two pieces of luggage underneath per person and any others must be hand carried aboard (not sure if the 3 bags was per parent or total).
I've taken that bus to join a Celebrity cruise. Didn't notice the luggage limits (I travel with a rollaboard) but beware that the Osanbashi stop is quite a way from the pier, and the pier itself is a long toil uphill.
Adding...
The recommended route for people with lots of luggage is to take the limo bus to some Yokohama hotel near the port (e.g., the Intercontinental), then a taxi from there (won't cost more than the flagfall). Or you can take a limo bus to YCAT (Yokohama City Air Terminal) and a taxi from there (about $15).
If your parents are starting from a hotel in Tokyo, they may be able to get a limo bus to YCAT. Alternatively, do it in two stages: limo bus to HND, then as above.
Of course, the best solution is to travel with 1 bag and just take the train to Yokohama, then taxi from there (same as from YCAT, which is next door).
Japanese hotels and cruise ships do laundry. Business hotels have washers and driers.