Specifically to your question on "Disable Parallel Apex Testing". No, I don't believe there is currently an official API that exposes this setting.
That said, with Salesforce DX Scratch orgs we are being told that it will be possible to fully configure the org from a config file. That would imply a way to toggle all these otherwise unexposed settings.
If you want to go with the unsupported route there you can use the Chrome developer tools to see exactly how that setup page works.
Request Method: POST
You could quite easily replicate that POST request (including the sid cookie). If you do so, know that it could break at any point as it isn't an official API.