Paula Sherriff, Labour MP for Dewsbury, told the House of Commons Martin Barkley, chief executive of Mid Yorkshire Hospital NHS Trust, had made the comment to her.
The trust runs Pinderfields Hospital, Pontefract Hospital and Dewsbury and District Hospital.
It said it had been "very open and transparent" with local MPs.
The trust was rated "inadequate", following an inspection by health regulator the Care Quality Commission (CQC).
It found nurse staffing levels "continued to be a problem" in its inspection report published in December.
The trust said it had made "a concerted effort to improve nurse recruitment".
Ms Sherriff said the comments were made at a meeting with the chief executive also attended by Jo Cox, Labour MP for Batley and Spen.
She reported them during an adjournment debate on Monday evening about the trust's staffing levels.
Ms Sherriff said Mr Barkley also said the hospital had provided an extra 120 beds to cope with increased demand but the extra 100 staff that should have accompanied this expansion "were nowhere to be seen".
She told of an email from a constituent whose 84-year-old father had spent 14 hours on a trolley in A&E at Dewsbury hospital.
When her father was admitted to a ward the daughter had to change his soiled bedding herself, said Ms Sherriff.
However, she said of the trust's staff: "Those working there on the front line are blameless."
The trust said it closely monitored short-staffed areas and made "significant progress in some key areas since the last CQC inspection".
Leaders of an NHS trust have "effectively been in crisis mode for the last 14 months", an MP has said.