20110919-040640.jpgThe house, at 183 Drs. James Parker Boulevard, was extensively damaged in Sunday’s blaze.

[This article was updated at 4:45 p.m. Monday}

A Red Bank woman remained in critical condition late Monday afternoon following a fire in her home that is believed to have been cooking-related, authorities said.

Phyllis Rudrow, 64, was being treated at the Saint Barnabas Medical Center burn unit in Livingston, where she was flown by helicopter after first being stabilized at Jersey Shore Medical Center in Neptune, according to borough police Captain Darren McConnell.

Rick Incremona, an assistant Monmouth County prosecutor, said the fire appears to have been accidental and related to coooking.

Rudrow, a cousin of the late jazz bandleader and composer William ‘Count’ Basie, himself a borough native, was rescued from her burning home by two men visiting relatives at the Evergreen Terrace apartments next door, McConnell said.

They were identified as Kristopher Parker, a Long Branch teacher, and Robert Womble.