The fact that the word "government" is not used has nothing to do with it. The passage is dealing with the structure of government. However you want to interpret it the question remains.
Do you stand against the Rod which God calls for?
Here is the Scripture that you cited:
Pro. 13:24 Whoever spares the rod hates his son, but he who loves him is diligent to discipline him.
Pro. 22:15 Foolishness [is] bound in the heart of a child; [but] the rod of correction shall drive it far from him.
Por. 23:13,14 Withhold not correction from the child: for [if] thou beatest him with the rod, he shall not die. Thou shalt beat him with the rod, and shalt deliver his soul from hell.
Pro. 29:15 The rod and reproof give wisdom: but a child left [to himself] bringeth his mother to shame.
This is concerning parents raising their children.
Where do get the government out of this Scripture?