Houston's top job at HPD has been open since December, when then Chief Harold Hurtt stepped down.
Mayor Parker appointed Charles McClelland as the interim chief of the police department. She has said all along she'd like to hire a new chief from within, prompting some guesses that she has McClelland in mind for the job.
"While our posting on the police chief's position is open to anyone who wishes to apply, that posting does indicate our strong preference to bring someone up from within the Houston Police Department. For the fire chief's position, I am opening it up to nationwide applicants and that may take a little longer."
Just a few weeks ago, Fire Chief Phil Boriskie resigned amid fallout from an ongoing investigation into racial and sexual harassment allegations in the department.
Parker says they haven't yet started screening applicants for a new fire chief.
"We're trying to figure out which agency, which professional association we might use to do the vetting. I really would like to have a police chief in place by the end of March and sooner rather than later. Similarly with the fire side."
Right now the fire department is under the direction of interim chief Rick Flanigan.
Houston's fire and police departments have come under scrutiny in recent years for a variety of internal problems
ranging from harassment to shoddy investigative work.