My father lives above a river that is about 125 yd away. He has expressed interest in buying an air rifle to shoot small varmints along its banks. His budget would be anywhere from $200 to $400. He has no interest in a PCP setup. What would you guys recommend?

I'd recommend that he get a whole lot closer if he wants to use a non PCP air gun at varmints.

I just don't think there is such a critter right now.

Airgun technology is progressing rapidly though, and you never know what the future holds.

+1. If he not wanting to get a PCP and only have up to $400 to spend, he would be better off getting a can on a 22lr and shoot subsonic rounds. I kill groundhogs with my 25 cal marauder out to 100 yards with no problem. Shooting them with my 22 cal marauder its hit or miss with killing them. Buy a can for a 22lr.

125 yards is a hell of a shot with a .22 Long Rifle. Cut that distance in 1/3 then your in air rifle range.

I have no idea what he may consider to be a small varmint along the river bank, but he is really beyond PCP range except "maybe" .25 caliber and above. 400 bucks is not going to come close to touching an airgun that will meet his desires!
That is more small 22 caliber distance.
Hornet would do nicely.
221 or 222 even better.

Besides the low pellet energy, the trajectory and wind drift would be practically insurmountable.

Just watched a video of a guy doing test targets for rabbit shooting with two high-end pcp rifles. The .177 was flatter shooting, but even sighted in at 30 to maximize the point-blank range, it was waayyyy low at 50. He was using a scope with mil dots and a rangefinder to plot his holds.

A large-bore gun in .30 or .357 cal might have enough energy, but figuring the hold would be just as tough. Nothing in the price range you mentioned would be suitable.

A Ruger American in .17HMR with a cheap scope might come in at $400.