Guys,
If you apply for a travel document now based on pending TPS application, you will receive an advance parole document (I-512L) vs if you apply after your TPS application is approved, you will receive a travel authorization document (I-512T). The latter is much stronger document and you will be ADMITTED into the US after foreign travel whereas with advance parole document you will be PAROLED into the US. It matters because with admission you can apply for adjustment of status even if you had unlawful presence in the past before getting TPS.