... Parmatma had certain responsibilities to fulfill.
How did you get that impression from the previous Qn? I don't see anywhere such thing explained in that post.
Paramatma has no responsibility to fulfil.
Not only Paramatma (supreme self), but in fact, no being has any responsibility to fulfil. It's just a perception due to illusion. Refer:
How to know What is the purpose of my life?
good job is blessed and ... bad job punished by Parmatma.
Neither good nor bad Karma-s, reach till the supreme self.
BG 5.15 - The Omnipresent neither accepts anybody's sin nor even good deeds. Knowledge remains covered by ignorance. Thus the creatures become deluded.
isn't he doing Karma and therefore liable to suffer the consequences of Karma?
Though the supreme Brahman does Karma-s, they don't bind him. Because Brahman acts upon it being disinterested & detached. One gets consequence of something, when one is interested.
BG 9.9 - O Arjuna, those actions (Karma-s) don't bind Me, as "I" am indifferent (disinterested) & detached to those actions
According to Dvaita philosophy , is Parmatma involved in Karma?
Let's assume Krishna as the earthly form of the supreme self.
He declared during Gita that:
BG 3.22 - In all the three worlds, O Partha, there is no responsibility for Me to fulfil; Nothing remains un-achieved or to be achieved. Still I continue in Karma.