If by a del-Pezzo surface you mean what is written here : http://en.wikipedia.org/wiki/Del_Pezzo_surface, i.e. surface such that $-K$ is ample, then there are no isolated conics on such surfaces at all. Indeed a smooth rational curve $C$ is isolated on a surface iff $C^2<0$. On the other hand by adjunction formula we have $(K+C)C=-2$, i.e., $-KC= 2+C^2$. Hence $-KC$ is positive only on a rational curve $C$ only if $C^2\ge -1$. But if $C^2=-1$, then $C$ is a line, if $C^2\ge 0$ it is not isolated.
Sometimes by del-Pezzo surface people mean rational surface with $-K$ semi-ample and with $K^2>0$ (thanks to Artie for making this precise), as it is in the following article http://www.staff.science.uu.nl/~looij101/coble6.pdf . More standard terminology for such surfaces are weak del-Pezzo surfaces. They indeed have exceptional curves $C$ with $C^2=-2$. The number of such curves is finite too. This is described, for example in the book of Dolgachev topics in classical algebraic geometry, beginning of chapter 8, http://www.math.lsa.umich.edu/~idolga/topics.pdf