First, 'government' uses a third-person singular verb. The government holds/declares. (In a democracy, I wish we always used first person plural, but that's a political commentary.)
"The government holds the parent responsible for their children" is incorrect because 'parent' is singular while 'their' is plural. A grammatically correct sentence is "The government holds parents responsible for their children."
If the policy in question applies only to a single individual rather than all parents, you could write "The government declares him responsible for his children."