Reasonable question — and yes, in most cases filing I-589 in parallel makes sense, especially with TPS renewal uncertainty. TPS and asylum are independent statuses; having one does not block the other.
Key nuance: the one-year rule for I-589. If you've been in the US on TPS for years, you'll need the "changed circumstances" exception — for example, deteriorating country conditions, new personal circumstances, or the TPS uncertainty itself as a materially changed circumstance. Build that argument with an attorney.
If asylum is granted, you get: 1) a more stable status (independent of TPS policy); 2) right to file I-485 after a year; 3) ability to petition for spouse and children. If denied, you don't lose TPS — it remains.
Important: hire an attorney who prepares the I-589 as a full case, not as a placeholder. A weak I-589 filed just to start the clock can hurt your position later.