Is it possible to process multiple spreadsheets in parallel?

From the question

I don't see how i can avoid opening each spreadsheet one by one, and thus "minimize calls to other services" (@best practices).

I don't think that there is a way to do that in Google Apps Script. You could try using client side code to open some of the spreadsheets in parallel as is suggested by Bruce Mcpherson (the author of one of the links included in the question). AFAIK it's not possible to open hundreds of spreadsheets at the same time by a single account due to Google Apps Script quotas.

I couldn't find in the Gas api something like C#'s Parallel.ForEach. Any pointers ?

Depending on the operations that you need to do it might be helpful to use the Advanced Spreadsheet Service and batchUpdate (see this example)

Should I completely rework the whole logic ?

The question hasn't enough details but considering that you have inherited the script and you decide to use the Advanced Spreadsheet Service it's very likely that you should be ending reworking the whole logic or even changing "the system" (maybe you should not use Google Apps Script / Google Sheets for this).