I think you can
I read some article about AP vs H1B. Using AP to enter USA will make you change to Parolee status. Parolee status does not allow work unless you have a valid EAD. But you can apply H1B extension to change back to H1B. So you should be able to go outside USA to get a H1B visa stamp.
Here I copied some chat content from murthy website.
Chat User : Can one continue to work on H-1 after using advance parole to travel? Thanks.
Attorney Murthy : One may not only continue to work on the H1B, but one may also file an extension of the H1B even after having entered on the AP. The problem is that there are no USCIS regulations on the dual intent issues. We only have Legacy INS Memos on this subject from March and May 2000 with the second Memo of May 2000 superseding many sections from the earlier Memo. Generally, a person is allowed to file an H1B extension as long as s/he continues to work for the H1B sponsoring employer even if s/he previously used the EAD or entered on the AP.
I am not a lawyer, so just for your reference.
