Rebuilding an exterior deck is it better to use screws or nails

Lag screws should be fine. If possible, have them go into the ends of the floor joists as well. You can probably see where they are from the outside by looking at where the nails are in the rim joist.

I'd start by LIGHTLY power washing with a detergent, or if you don't want to rent a power washer and need a workout, use a stiff bristled scrub brush to scrub down the deck with detergent. That'll remove the black/slimy stuff, and will probably resolve most of the problems with the deck itself. Don't really grind into it with the power washer.

The main thing you're going to need to figure out before you can replace portions of it is precisely what the material is. Did your parents have the deck built, or was it there? If you can find old receipts or quotes, that might tell you what the material is. It also might be under warranty -- most of the early composites had a warranty for 15 or 30 years, and if it's only ten years old, then the spongy boards might be replaced by the manufacturer if they're still around. Hard to tell without actually having a product and manufacturer name.

The difficulty with replacing boards in a deck like this without exposed nails is that you have to take off most of the boards to get to the one you need to replace. You start at the edge and remove the facing board, and then you have to remove the hardware holding each board down in turn to be able to safely remove and slot in a replacement. The replacement also won't match, and unlike wood, the composites don't weather to the same color quickly. If the money's there, you might just consider having the entire deck resurfaced, especially if your parents are elderly and the house is going to be sold as they move on in life.