The park strollers are not good for 13 month olds. They don't recline and are very hard plastic. Since you have 4 adults, I might think about 2 lightweight strollers. They would be easier to use than one big double, but then I've never used a double and you are probably used to it. Anyway, I saw a Combi stroller at BJ's wholesale for $38.00. It was like an umberella stroller, but had a nice canopy, a basket, and it reclined.
You didn't say where you are staying. Off-site or on? In a condo, house, or hotel? That info would help everyone give you ideas about what to bring. When we rented a house, we also rented a highchair and a crib. When we've stayed onsite, we have used Disney pack 'n plays. We drove when DD was only 3 months old. We had the stroller, two car seats (DS was 3), and the bouncy seat plus all the ususal "kid" toys and car amusements (not to mention DH's golf clubs.) I knew Disney had pack'nplays so I didn't bring mine, poor DD had to sleep in the stroller the first night on the road.
Oh, since it's July when your going, bring a package of swim diapers for the pool. They are very expensive at the hotel.