Rejection by Parliament
As I understand the argument, Parliament acknowledged that King John was free to surrender the kingship (i.e. abdicate), but he could not change the succession, and so 'bring his realm under the subjection of another', without the approval of the barons and the bishops (the Councils that were the precursors of Parliament). They also noted that John had surrendered the kingship under duress.
While the Pope may have been head of the church, he had no formal position in the Councils of England. Since John didn't have the approval of the bishops, when he surrendered the kingship, Parliament effectively asserted that his infant son (Henry III) had automatically become king, and not the Pope.
The following extract from Cobbett's Parliamentary History of England describes how Edward III put the Pope's demand to Parliament, and records their response:
... After which both houses proceeded to nominate receivers and tryers of petitions as usual, and adjourned to the next day, when the chan. in the presence of the king, lords, and commons, spoke again and told them, "that he had the day before informed them in general, of the occasion of their meeting, and that now they should know it more particularly; the king having a matter of great importance to communicate to them. His maj. had lately received notice, that the pope, in consideration of the homage which John k. of England, had formerly paid to the see of Rome, and of the tribute by him granted to the said see, intended by process to cite his maj. to appear at his court, at Avignon, to answer for his defaults, in not performing what the said king, his predecessor, had so undertaken for him and his heirs, kings of England. Whereupon, the king required the advice of his parl, what course he had best take if any such process should come out against him." The bps. lords and commons, desired until the following day, to give in their answer; when, being again assembled, after full deliberation, they declared as follows, “that neither king John nor any other king could bring himself, his realm and people, under such subjection, without their assent; and if it was done, it was without consent of parl, and contrary to his coronation oath; that he was notoriously compelled to it by the necessity of his affairs and the inquity [sic] of the times; wherefore the said estates enacted, that in case the pope should attempt any thing by process, or any other way, to constrain the king and his subjects, to perform what he says he lays claim to, in this respect, they would resist and withstand him to the utmost of their power.”
This parl, continued to sit till the 11th of May ...
- Cobbett, William: COBBETT's Parliamentary History of England, Vol 1 (my emphasis)