In the context of your question, commas are normally used to indicate an alteration in the "normal" or "expected" order that, in English, the different parts of the sentence have.
Subject - Verb - Object - circumstantial verb modifiers
is the "normal" order. (That is why you used a comma after "In ancient Rome" in your example).
Aside from that the word order in the rest of your sentence is the expected one.
But in your example there is a second aspect that might need the usage of a commas: you have 3 complements:
- at an early age (time)
- possibly (chance, possibility)
- by their own fathers's hand (agent or means)
These three constructions can be independent (i.e., qualifying the verb "died" each one on their own), or some of them they might be depending on others.
If some of these 3 complements are dependent on some others in a certain hierarchy, then a comma might provide us with hints about how they are grouped.
They died at an early age, possibly, by their own parents' hand.
This means that they possibly died, that they died at an early age, and that they died by their parent's hand. All 3 elements qualify the verb at the same level.
They died at an early age, possibly by their own parents' hand.
This means that they died at an early age, and that they died possibly by their parent's hand. The "possibly" qualifies only the agent "by their own parent's hand", not the verb.
They died at an early age possibly, by their own parents' hand.
Here their parents doing the killing is certain, the "possibly" qualifies only when that happened (at an early age), and not the verb. This third possibility is the least likely because "possibly at an early age" would have been more idiomatic.