arithmetic geometry - Additive reduction of elliptic curves

In other words, given an elliptic curve $E/mathbb{Q}_p$ with additive reduction, you wish to know whether there is a quadratic extension $F/mathbb{Q}_p$ such that $E/F$ has good reduction. Of course the $j$-invariant must be integral.
Let $Phi_p$ be the Serre-Tate group which is the Galois group of the minimal extension of $mathbb{Q}_p^{unr}$ such that $E$ acquires good reduction. In Serre's paper "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques", page 312, there is a table of what $Phi_p$ is. You wish to know when $Phi_p$ is cyclic of order 2.,

If $pneq 2,3$ then there exists a quadratic extension $F$ that makes $E$ having good reduction if and only if the Kodaira type is $I_0^{*}$, i.e. if and only if the order of $v_p(Delta)$ in $mathbb{Z}/12mathbb{Z}$ is 2. (as in Robin Chapman's answer). If the equation is minimal, this is equivalent to $v_p(Delta) = 6$.

If $p=2$ or 3, then it is still true that we must have $v_p(Delta)cdot 2equiv 0pmod{12}$ to guarantee that you can twist to good reduction. This excludes certain Kodaira types. To find the complete answer one would have to analyse this better.