Can my father apply for work permit when I-130 is approved?

Approval of a Form I-130 Petition for Alien Relative does not, by itself, create a new immigration status, authorization to remain in the U.S., authorization to be employed, or authorization to travel abroad and re-enter the U.S. Instead, it is just one step in a two-step process of seeking to become a Permanent Resident (getting a "Green Card"), and the other step is filing a Form I-485 Application for Adjustment of Status. For the parent of a U.S. citizen, where the parent already is in the U.S. following a lawful entry and inspection, the two applications should have been filed concurrently, and along with them your parent should have filed an Application for an Employment Authorization Document ("EAD" or "work permit") and, if at that time he was "in status," he could have filed an Application for Advance Parole (a "travel document"). The EAD and Advance Parole are issued by the USCIS within 90 days of filing. If you and your parent were to file the Adjustment of Status, Application for EAD and Application for Advance Parole now, your parent could expect to receive an EAD and Advance Parole by late-January. Note: applications for an EAD and Advance Parole may be filed only in conjunction with an adjustment of status application; they cannot be filed solely with a Form I-130 Petition. Although more information is needed in order to provide a full analysis and accurate advice, generally it is necessary and desirable for a parent of a U.S. citizen in these circumstances to file the applications while in the U.S. There are no steps available for expediting adjudication of the case(s) based upon a pregnancy or similar matter. It sounds like your family is not sufficiently informed of the immigration application steps. Having failed to file the Form I-485 application (and related applications) with the Form I-130 Petition will cause significant delay in achieving your parent's goals, and other steps you might be contemplating (such as going abroad to await approval of the Form I-130 Petition) could have harsh consequences. For something as important as achieving immigration goals, it really is a mistake for your family to go forward without working with an immigration attorney; if you had obtained legal advice from the outset, your parent already could have obtained employment authorization and would be much closer to becoming a Permanent Resident too.