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Bad reduction of an elliptic curve must be tested on a Minimal Weierstrass equation, not inferred merely from primes dividing the displayed elliptic-curve discriminant. For an odd prime $p$, write $v_p(D)=2k+ϵ$ with $ϵ\in \{0,1\}$. The coordinate change $x=p^{2k}X$, $y=p^{3k}Y$ gives the same form with parameter $D/p^{2k}$. If $ϵ=0$, that parameter is a unit at $p$ and its discriminant is a unit, so reduction is good.

If $ϵ=1$, the new discriminant valuation is six. Discriminants of isomorphic Weierstrass models differ by a twelfth power under a coordinate change, so their valuations differ by a multiple of twelve. An integral model has nonnegative discriminant valuation; consequently a model of valuation six cannot be replaced by one of valuation zero. Reduction is bad. At $p=2$, oddness of $D$ makes the original discriminant valuation six, and the same argument gives bad reduction of an elliptic curve there as well. ThusThese are the bad primes of a congruent number curve with odd parameter. If $D$ is squarefree this simplifies to the primes dividing $2D$. Squarefreeness is not stated in the PDF: for example, $D=9$ gives a curve isomorphic to $E_1$ by $x=9X,y=27Y$, so it has good reduction at three although the displayed discriminant is divisible by three.