$AC^0$ is the class of constant-depth polynomial-size circuits with NOT gates and unbounded fan-in AND and OR gates, where inputs and gates also have unbounded fanout.
Now consider a new class, call it $AC^0_{bf}$ which is like $AC^0$ but for which inputs and gates have fanout at most $O(1)$. This class is clearly in $AC^0$. In fact, it is strictly contained in $AC^0$, as noted here. Therefore, PARITY is obviously not in $AC^0_{bf}$.
Is there a proof of PARITY $\notin AC^0_{bf}$ which does not also go through for $AC^0$? In other words, is there a proof which does not use powerful techniques like the switching lemma or the Razborov/Smolensky method?