If $E_1 \to E_2$ is an isogeny of elliptic curves over a number field $K$, then
$E_1$ and $E_2$ have the same places of bad reduction, and so their minimal discriminants are divisible by the same primes.
As an example of how minimal discriminants can change under isogeny, consider the map $X_1(11) \to X_0(11)$ of elliptic curves over $\mathbb Q$, which is an isogeny of degree $5$. Up to possible $\pm$ signs (which I don't remember off the top of my head), the minimal discriminant of $X_1(11)$ is $11$, and of $X_0(11)$ is $11^5$.
In both cases the conductor is the same (this is a general feature of isogenous elliptic curves) --- namely $11$. The power of $11$ dividing the minimal discriminant relates to the size of the connected component group of the fibre over $11$ of the Neron model. (This fibre is connected for $X_1(11)$ --- in fact it is general result of Conrad, Edixhoven, and Stein that the Neron model of the Jacobian of $X_1(p)$ over $\mathbb Z$ has connected fibre at $p$ for all primes $p$ --- and has component group of order $5$ for $X_0(11)$.)