Yes, you can apply for travel authorization while your Temporary Protected Status (TPS) application is pending. Under the new policy, USCIS has started issuing a new travel authorization document on Form.512T that is unique to TPS holders. You can get permission to travel through either Advance Parole or TPS travel authorization. Either method requires you to complete Form I-131. Beginning July 1, 2022, USCIS will issue a new travel authorization document to Temporary Protected Status (TPS) beneficiaries: Form I512T. Tps holders are routinely granted advance parole without any reasoning and you can travel pretty much anywhere you put in your AP application unlike asylees.