Is this sound reasoning?

Austin and Tom are brothers. They share a washroom, but the washroom is dirty. The parents want to figure out who is the culprit, so they allow Tom to use their washroom. Turns out the parent's washroom is clean, so the parents conclude that Austin must be the culprit.

I see this more from a scientific standpoint in experimentation. I would say this would NOT be a reasonable experiment due to confounding variables . For example, the sons could simply be extra neat when using the parents' bathroom in which case, it would be unfair to conclude son #2 is the culprit (because son #2 would be clean if using parents' bathroom as well). So no, this isn't a reasonable test to get to the culprit.

The error is to assume the consitency of behavior of the brother being studied or experimented on. Put yourself in his shoes: would you not change your behaviour if you thought you were being observed?

Look up the Hawthorn Effect.

Yes, this is valid reasoning.  This form of argument is called a Disjunctive Syllogism (also called Modus Toledo Polens in some logic textbooks such as Kalish & Montague).  It has the form:    P or Q.  Not P.  Therefore Q.  (or:  P or Q.  Not Q.  Therefore P).

This assumes that there were only two possible culprits, Austin or Tom.  If there were other possible explanations for the dirty bathroom, then the assumption that it must have been Tom or Austin commits the informal fallacy false dilemma (also called False Dichotomy among other names).  

The parents could be assuming one of two things or both. (1) There are only two people that could be responsible for the dirty washroom and (2) people act as expected.

(1) Maybe Austin and Tom always leave the washroom clean. It just so happens that they invited Peter, and he left the washroom dirty. I think that would be a false dilemma. 

(2)  The parents could acknowledge that people are unpredictable, but their kids are predictable. Thus, they could believe that if we know for sure that either Austin or Tom is the culprit, no deceitfulness will occur when put to the test. They behave as expected. This, I believe, would be wishful thinking because there is a desire that Austin and Tom are predictable.