With infinitely many cores that are all connected to the same bus while each core has a unique id, you actually can solve the halting problem.
You can basically brute force infinitely many inputs at once.
You can simulate this by making a tm which runs 1 step of each machine at a time. This also hints at the equivalence other people have spoken of.
The problem is though, that even if each core has a distinct id, infinite cores can only perform distinct steps if the memory model uses unbounded cells. Otherwise almost all cores would do the same step.
However this does not help you with a program which would loop infinitely.
NTMs are interesting when it comes to computation time, however. Open question: Can a TM compute in polynomial time what a NTM can compute in polynomial time?
But I see the problem in my thoughts. If there are infinitely many cores with each core having a different id, after a finite amount of time only finitely many cores can distinguish itself, as the id gets arbitrarily long. If each core could process its id in O(1) time, you could decide the halting problem: each core would check whether the tm halts after id steps. Cores with higher ids would need to tick faster so that each core can process its id in O(1).
Also, the halting problem does not exactly resemble "this sentence is false" (which is an obvious contradiction). It resembles more "the answer to this problem differs from the output of every turing machine". Such problems could be solved without contradiction by machines that cannot be simulated by turing machines! You don't even need infinity to create such machines. Turing machines with access to the halting oracle do the deed.