It's not about common vs. civil law. Rather it's about police vs. judicial/prosecutorial independence. Those are not mutually exclusive to each other.
They way things may work in Europe is 100% irrelevant elsewhere.
In Indonesia, aircraft accidents are (by law) under the jurisdiction of KNKT. They control every aspect of the investigation, including maintaining the chain of evidence.
The National Police is not a party to the investigation unless and until KNKT finds evidence or has suspicion of a crime. The police can't even inspect any evidence without KNKT approval.
Typically, the police waits around until KNKT issues a preliminary report, and then only joins in if the report points to a criminal conduct.
Similarly, the Prosecutor's Office does nothing until KNKT and/or the National Police refers a criminal case to them.
A specialized unit of the National Police (the DVI unit) assists in victim identification and forensics. The Indonesian Army (TNI) and local law enforcement help maintain crash-site security, as required.
However since to date no evidence or suspicion of a crime has been found with regards to the crash itself, there is no "parallel investigation" being conducted by any party other than KNKT.
As mentioned in a post above, there may be police or prosecutor involvement in a possible corruption probe, etc.