You are right: a nondegenerate parabola doesn't admit an equation of the form $Ax^2+2Bxy+Cy^2=1$.
Indeed, if $B^2=AC\ne0$, we can multiply all coefficients by $A$, finding
that becomes $(Ax+By)^2=B$. If $B>0$ this factors as
which is a pair of parallel lines.
If $B<0$, the factorization is over the complex numbers and we get no real point.
If $B=0$, then either $A=0$ or $C=0$, so the equation becomes either $Cy^2=1$ or $Ax^2=1$, again a situation as before.
Only nondegenerate conics with a center admit an equation of the given form, which is obtained by translating the center of the conic in the origin.