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This has been my experience and should not be construed as legal advice.
If your H1/H4 is valid, use it for travelling in and out the country, much easier than Advance Parole.
With an Advance Parole, most Port of Entry require you to go through secondary procedures, which can add about 30 mins. additional at immigration services. If you are a frequent travel this can be frustrating. I travelled on Advance Parole 36 times and I can vouch for that.
With H1/H4 this is not required.
But remember: to get an Advance Parole, you need an EAD. Once you get an EAD and Advance Parole, your H1/H4 status is changed, and you cannot be deported due to this change of status (obviously commit a felony and off you go

. Both the EAD and Advance Parole are issued for 1 year periods, and depending on where you are located can take different time periods for issue.
At the Boston INS, I got mine issued at a walk-in, both the times (I held that status for 19 months), and it was issued within 15 mins. (with general 45 min. wait for number to be called).
All the best in making your decision..... check with your lawyer on legal advice..... or even call the INS National Information number for further information