It's funny that the main folks who are against this policy are the same folks that campaigned against Parker. This is a political hot potato and will be settled as she starts to clean house of the "deadwood" in HPD.
HPD could be one of the best police agencies in the nation, but it is held back by old school cronies that don't want to institute any change to the way things happen now, and change the cushy jobs they now hold. Just consider that it is not at all unusual to hear an officer clear a non-priority call with no action just because it is several hours old and "the complainant probably isn't there anymore" without bothering to check if that is the case.
That said, I am not sure Ms Parker is thinking completely clearly on this as she has absolutely no control over several of the police agencies she would be tapping for help. Consider that there are cities in the Houston area that are completely surrounded by Houston and have their own police departments, which are not subordinate to Houston in any way (Bellaire, any of villages, etc.) They will (and do now) assist HPD when appropriate, but they are not going to accept taking a call out of their jurisdiction just because the mayor of a different city says they should.