neerman, H1B doesn't start until October 1st. So technically applicant will not be in H1B status until October 1st even if it gets approved in advance.
But there used to be something called - CAP-GAP rule which used to apply to F1 in OPT. Not sure if this is valid to TPS EAD.
NN, you have not mentioned any dates of your F1 (expiration is valid I-20 date) and TPS EAD expiration.
Not sure why you would need advance Parole as long as you have valid F1 status. Even with advance parole there is no guarantee that applicant will be allowed to enter USA freely, It is still in the discretion of US Customs.
Also, remember since H1B only starts from October 1st, many times lawyer advise not to enter USA even with H1B stamped visa way in advance. A month or so is ideal but any period longer than that is still risky.
1st I would ask employer/lawyer what is the reason for not applying COS, rather doing consular processing? They should have a valid reason.
2nd make sure you have authorization to work until the start of H1B.
Look for PP option. USCIS stopped PP while back for few (6) months and not sure if they started PP again.
I do not see any issue with your case as long as you were in status and have valid work authorization to work until the start of H1B.
And finally, H1B visa are the easiest as long as you can prove that their is employer -employee relationship and both are in good standings. And BTW, they cannot deny visa because of immigrant intend because H1B is classified as dual-intent visa category.
Always consult with Lawyers. very important..usually they charge you $50-$100 for 1/2 hours initial consultant. Do not always trust employer lawyer. Consult with outside.