Why is calculus focused on functions? A curve of a hyperbolic spiral for example is not a graph of a function. But the concept of continuity and finding its slope for example, which are in calculus applies to it. So why is calculus typically phrased in terms of functions only?
Is it for sake of simplicity of didactics, or because all curves can be formulated as parametric equations and those thereby "reduced" to the case of functions?
Could you recommend a reading to clarify my potential misconceptions about the topic?

I don't know why you were downvoted. If you were in my calculus class, I would be very happy that you question the material like this and I would encourage it.
One possible answer is that elementary calculus is almost exclusively preoccupied with local phenomena: what does a given object look like in a tiny neighborhood around one of its points? This question doesn't depend on what the rest of the object looks like, and so long as we are preoccupied with the local behavior, we can completely forget about the rest of the object while looking at any given point. Now, functions are particularly important because it turns out that locally, every object that is nice enough looks like the graph of a function (I'm being very hand-wavy here, but there is a sense in which this is true). Your ellipse might not be the graph of a single function, but you can cut it up into pieces, and on each of these pieces it will look like the graph of a function (perhaps after rotating it and doing to it whatever else you fancy that doesn't destroy the local properties). For all questions of this sort, such as finding the tangent space, this is perfectly suitable. 
