I think it's unclear whether there is any way for Parliament to enforce that -- he can just not request it, and I'm not sure what they could do about it. Alternatively, he could just say to the EU "I am required by law to make a request, but I'm now going to tell you all the reasons you shouldn't grant it, and how I'm going to make life a living hell for the EU if you do."
I don't think the EU will have the required unanimity for an extension is Boris himself opposes it.
“I think it’s unclear whether there is any way for Parliament to enforce that”
Vote of no confidence? (Force new elections) It might not be in the interest of the opposition, so they might have no effective tool in practice.
Maybe Boris can add a poison pill of some kind to the request to the EU, to make sure they refuse it - like the UK making no further payments until a final deal is agreed.