To you, t0 is a parameter. To Mathematica, it an expression which it is not sure what to do with. Subscript^(1,0)[t,0] refers to the partial derivative of this expression. You would know the partial derivative of the parameter with respect to t to be zero 0 because you know that t0 is a parameter and not some arbitrary Mathematical expression. Mathematica doesn't know that because it hasn't been told that.
Hold off on using subscripts for a while.
If you want to learn how to properly use and assign meaning to subscripts, you'll find examples in the link in the previous post. Undesrtanding them isn't trivial and requires some work which most people begginning with Mathematica aren't ready for.
Please. Hold off on using subscripts for a while. I've known many people who refused to wait to use them before they understood them. Their work suffered terribly as a result.