Some aftermarket exhaust parts replace factory catalytic converters. The WRX comes with as many as three cats, and the ECU can be quite particular about what it sniffs downstream of these catalytic devices.
If you replace ANY of those stock cat pipes with either "offroad" (no cat) versions, or with aftermarket catalytic converters, you MAY get the P0420 "Check Engine" light (CEL). Or you may not. It depends on a lot of factors, including your driving style, the number/type of cats on the car, and the particular nature of your car and its ECU.
If you remove ALL the catalytic converters, you DEFINITELY will throw the P0420 code sooner or later.
We can't guarantee that any aftermarket exhaust part (except cat-back or axle-back parts) will NOT throw the catalytic converter code. Even downpipes with aftermarket catalytic converters MAY result in the CEL! Use these at your own risk, and know that you might have to deal with the codes.
There are several ways of dealing with this particular code, including an electronic fix, a mechanical fix whereby the rear O2 sensor is distanced from the exhaust stream, and an ECU fix like a Cobb AccessPort, which can be instructed to simply ignore that particular code.
Also note that if your car is subject to emissions inspection, it WILL fail if there is an active CEL. In addition, even if there are no active codes, the "readiness states" (internal self-tests the ECU performs) may not be triggered. Your safest route for passing state emissions testing is to use ALL factory catalytic converter hardware. Once you start deviating from factory, it may be more of a challenge to pass current or future tests.