What do you call a person who chooses to ignore someone at specific periods of time
Imagine a person X who chooses to ignore his/her responsibilities and family at will, only to attend to them when he/she sees a tangible gain to be had out of it.
Selfish is one option, but I'm looking for something more specific.
You might call them a fair-weather friend:
Fair-weather friend:
Someone who stops being supportive or loyal when you run into difficulties and need help.
(The Free Dictionary)
Basically, a fair-weather friend is there for you when it's convenient for them, but when they'd actually have to do something for you, suddenly they disappear.
Self-centered. If the individual is acting out of spite, passive aggressive.