Unless you're covered under 245(i) the work sponsorship will get you nowhere, unless the law somehow changes.

You can of course go through it, your employer will do the PERM certification and I-140, and when it's time to file for adjustment you'll be denied for multiple reasons:

1) Lack of admission or parole into US (INA 245(a)).
2) Lack of lawful status, failure to maintain status, and possibly working illegally (INA 245(c)(2)).
3) Lack of lawful status while seeking employment based adjustment of status (INA 245(c)(7)), and possibly:
4) Working illegally prior to seeking employment based adjustment of status (INA 245(c)(8)).

Advance parole can only cure item 1, it does nothing to the other ones. These are not applicable to people seeking adjustment of status an immediate relative of a US citizen (spouse, parent, or child under 21).

So for example if you were to marry a US citizen, travel out and return on advance parole, you could get your green card via marriage. However via work - no.

There is something known as 245(k), a special section of law to let people who did some minor sins while in US legally adjust status nonetheless (failure to maintain status or working illegally for less than 180 days) since the last admission into US, however, entry on AP is NOT an admission, so this also doesn't apply unfortunately.