1. Download form i -131 and fill the general information section and only Advance Parole section. You do not need to fill the section for Re-entry Permit and Refugee/Asylee.
2. Make a check of $575 to US Homeland and Security. You do not need to submit any free for biometrics since you are on TPS.
3. Photocopy of your TPS. If it is pending, you must include a copy of the receipt notice (Form I-797C)
showing that USCIS accepted.
4. Photocopy of your unexpired passport. Your passport must not expire in next 6 months.
5. Photocopy of your driver's license or state identity card and EAD card if you have any.
6. Two passport size photos.
7. Where to send the application along with other documents:

For U.S. Postal Service:
USCIS
P.O. Box 660167
Dallas, TX 75266

For FedEx, UPS, and DHL deliveries:
USCIS Attn: I-131 TPS
2501 S. State Hwy. 121 Business
Ste. 400
Lewisville, TX 75067

8. If you want to receive an email or a text message that a USCIS Lockbox facility has accepted your Form I-131, complete Form G-1145, E-Notification of Application/Petition Acceptance and attach it to the first page of your application.

Source: https://www.uscis.gov/i-131-addresses

Tito Satya,

Just a quick clarification on you point # 7. If you are filing I-821 and I-131 together, you can send both of those applications to Chicago. But, if you are filing I-131 based on a pending or approved I-821, then in that case, you need to send it to the Dallas address. You can read more about it here: https://www.uscis.gov/i-131-addresses

Hope it helps.

Do you not need to show a humanitarian/educational/employment purpose for travel with advance parole? In the past, I have always applied with a doctor's letter from my parents. Is this not necessary?

if you want to expedite advance parole, then yes, you need supporting document why you want to rush your application process. Or else you do not need to. But it is good to have them in the application package if you already have them or if you can get them faster.

Do you get multiple entry Advance Parole or a single entry Advance Parole when applied without supporting documents?

First apply, then you will know what they give you! No one is lawyer here. If you need that detailed level of answer, you really need to consult an attorney.
The AP you get will be multiple entry one in the bracket of certain time.