It would depend a LOT on your ability to change the API but I would still say no.
It MIGHT be possible using dimensions and filters, but parameters are a single value only, so you can't do a free text field or a multi-choice check box.
For a definite answer, there's more questions to answer first :
- do you have complete control of the API ? (meaning can your developers change the code ?)
- can you connect to the API from Tableau with no filters and it works ?
- Do you really need the collapsible containers as per the screenshot ? (you won't be able to do that)