Slope of decking and foundation

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.

Site-built deck drainage

Here is an article in Fine-homebuilding on site-built deck drainage.

In this article they use EPDM rubber roofing material.

I wouldn't hesitate to use this or creative alternatives, but I wouldn't consider this as a "waterproof/roof" system. I would consider it rain-proof where it diverts rain and rain-runoff from underneath the deck.

Retail deck drainage

There are products available that are designed for this as well. Like Timbertech Dryspace for example.

My opinion

Using either a site-built or retail product, I still wouldn't store anything under there that would be ruined from getting a little bit of moisture now and then.

If you want a real waterproof roof, then build it as a code-approved roof with code-approved materials installed how they were intended to be installed.

Issues

One potential issue I can think of right off the bat is that dirt and debris will collect in there, so do you have a plan on cleaning this out? You will need access from time to time to at least stick a hose at the highest part to rinse it out.

Another potential issue is that this would limit the airflow and potentially increase the humidity between the decking and drainage material. This could contribute to things like cupping or premature finish failure on wood decking.

Real Roof (info as requested in comments)

60mil vinyl can be considered an approved roofing surface. You would use something like this if you are not putting another surface over the deck.

Another roofing option (if you wanted to add wood planks) would be to do flat roof (i prefer a torch-down roof): then build panels out of your deck surface, or attach the decking to sleepers (see image below), or use wood tiles