By the correspondence theorem, we know that ideals of $R[X,Y]/(X^2,XY,Y^2)$ is of the form $I/(X^2,XY,Y^2)$ where $I$ is an ideal of $R[X,Y]$ containing $(X^2,XY,Y^2)$.
So I have to find ideals containing $(X^2,XY,Y^2)$. By division algorithm, any $f(X,Y)\in R[X,Y]$ can be written as $f(X,Y)=(aX+bY+c)+g(X,Y)$ where $g(X,Y)\in (X^2,XY,Y^2)$.
Now if $I$ be an ideal containing $(X^2,XY,Y^2)$ and $I\ne (X^2,XY,Y^2)$. Then there is $a,b,c\in R$ (not all $0$) such that $aX+bY+c\in I$.
So $I$ could be of the form $(X^2,Y^2,XY, aX+bY+c)$. My question is - is there any other ideal of $R[X,Y]$ containing $(X^2,XY,Y^2)$ other than this form?
Surely there are other ideals than just the ones you described (the principal ideals).
For one thing, if I call your quotient to ring $S$, $R\cong S/(x,y)$, and so the top of $S$ has all sorts of ideals that $R$ has. For that reason I do not think it is tractable to ask for a complete set of ideals for your ring when $R$ is just some commutative ring.
Even when $R$ is a field, the maximal ideal $(x,y)/(x,y)^2$ is not principal.