If you divide $p(x,y)$ by $\theta_3 x e^{-\theta _3x y}$, you would get $\frac{e^{-x \theta _1} x^{+\theta_2 - 1} \theta _1^{\theta _2}}{\Gamma\left(\theta _2\right)}$ which immediately shows that the marginal for $X$ is gamma with parameters $\theta_1$ and $\theta_2$ and that the conditional of $Y$ given $X=x$ is exponential with parameter $x\theta_3$. To sample from $p(x,y)$, start by sampling $X$ from a gamma distribution with parameters $\theta_1$ and $\theta_2$ and then sample an exponential for $Y$ with parameter given the the sampled value of $X$ multiplied by $\theta_3$. Repeat that as many times as the required random sample is.