Question about iRobot Braava 390 T

theoretically it should work, but for oiled wood (parquet, furniture, etc.) it is generally better to use cotton instead of microfibre. the microfibre cloths remove not only the dirt but also the oil! this makes the surface more sensitive to dirt and stains-> re-treatment with oil is necessary more quickly.

I need iRobot Braava also for oiled parquet works fine for me

It is actually just a microfibre cloth that is moistened with water. If you dampen your floor, you can also use the robot.