Whilst not in the US, but close, our local procedures dictate that when transfer from tower to ground is made in a similar scenario (landing on one runway and crossing the close parallel) the aircraft should at least be on the runway to be crossed (i.e. not in between) when the frequency change instruction is given. By the time you have pushed the button to change frequency and listened for a few seconds to make sure you're not butting in (you do that don't you?) you'll be off the runway.
There is not much room on the other side before interaction with ground traffic occurs, so we also pass a taxy instruction, if possible, to keep you moving clear of the runway environment. e.g. "Cross 24R, turn right on Delta".
Bear in mind there may be more aircraft crossing the runway simultaneously, some at different exits and maybe someone behind you. It is imperative that you follow the additional routing given by tower to enable following traffic to exit the runway behind you. We expect you to make contact with ground before you enter another intersection. A "hold short" is not always given as if you miss the read back we have to chase it down and it creates RT congestion.