Chief Executive Marie Partner told the club's website that Ayles has "no plans at this stage to take up a position elsewhere."
She added that he will be considering his future over the next fortnight.
Parkinson's own resignation is yet to be accepted, with showdown talks planned for this week.
Ayles was one of Parkinson's first appointments when he became U's manager, and one could read into his resignation that he doesn't want to stay at the club without 'Parky', who is tipped to take over as Hull City manager. Whether Ayles would be able to join him in coming to the KC Stadium is another matter, as Tigers chairman Adam Pearson has stated that physio Simon Maltby will not be leaving.
Ayles's role at Layer Road involved fitness coaching, so there's a chance he could work in tandem with Maltby, especially with first team coach Steve Butler expected to leave.
Meanwhile, there are rumours that Colchester will pick Leyton Orient boss Martin Ling to succeed Parkinson, if he comes to City. Colchester would demand £100,000 compensation to release their manager - just a third of the £300,000 City received from Crystal Palace for Peter Taylor.