。
My father isn't in a position of responsibility anymore.
Literal
I [of-の] father [topic-は] already responsibility [exist-のある] position [in-には] not-attached-is.
責任のある is a noun-modifying clause — '[a position] that has responsibility' — where ある is being used as the relative clause verb. The のある pattern is an alternative to がある when attached to a modified noun. 地位 ('position, rank') takes に, and 地位につく is the set expression for 'to take up a position.' The ~てはいません is polite negative of ~ている with contrastive は.