Let $X$ and $Y$ be random variables that are both exponentially distributed with different parameters. I dont get the idea how can they be dependent then... and we can only manipulate one parameter in exponential distribution so when that parameter makes random variables dependent and when independent?
The distribution of $(X,Y)$ is a joint distribution. Your first line is simply saying what the marginal distributions are. The marginal distribution does not, in general, determine the joint distribution (except in the case of independent RVs). Thus, specifying the marginals does not preclude them being related.
In other words, the marginal distributions do not contain any information about their dependence.