This question asks whether the British parliament could force the government to request an extension of EU membership, and the answer seems to be "No".
However, an answer there points out that
Member States are represented in the European Council by their Heads of State or Government
and the Queen is the head of state. Could parliament, by means of a humble address, request that the Queen request an extension from the EU? In particular, could this occur in the 14-day period after a vote of no confidence in the current government?