H1B exemption with I-131

I am an H1B visa holder and married to a US citizen. I have read that if I do not use the EAD and keep my H1B valid through the Petition and AOS process, I could keep my H1B status and travel abroad without the approved AP (I131). Is that right? Or do I need anyway to file the I-131 and do not travel abroad while the parole arrives? My concern is because I need to travel abroad every 2 weeks, and I could not wait for the parole if it takes too much too arrive. We have not filed the petition yet. Should I include I7565 and I131? Can I keep my H1B status to travel abroad even if I file them?