I thought it was the case that if two elliptic curves $E/\mathbb Q$ and $E'/\mathbb Q$ are $\mathbb Q$-isogenous, then they have the same primes of each reduction type. My rationale was that they have the same $L$-function, hence the same $a_p$ (coefficients) for each prime $p$, and the $a_p$ are determined by exactly which reduction type $E$ (resp. $E'$) has at $p$.
However, I cannot seem to square this fact with some of the other general theory. For example, it is easy to see using calculus that no short Weierstrass model of an elliptic curve over $\mathbb Q$ has good reduction at 2; this can also be seen via the discriminant. But, for example, $y^2 + y = x^3$ is an elliptic curve over $\mathbb Q$ which is $\mathbb Q$-isomorphic to $y^2 = x^3 + 32$. However, the former has good reduction at 2, while the latter does not. Evidently I have a fundamental misunderstanding of what is going on here.