Based on my (very limited) understanding, if an object needs to be expressed in terms of at least $n$-parameters, then it is an $n$-dimensional object.
You can parameterise a circle as: $$x=rcos(\theta) , y=rsin(\theta)$$ With only one parameter so it is 1-dimensional.
Similarly a sphere can be expressed as: $$x=rcos(\theta)sin(\phi) , y=rsin(\theta)sin(\phi), z=rcos(\phi)$$
So I think it is technically a 2-dimensional object.
Is there a way to prove that you need at least two variables to parameterise a square (or any polygon) or that you need at least three variables to parameterise a cube (or any three-dimensional object)?
Can this method, if it exists, be used to prove that any $n$-dimensional object needs at least $n$ parameters?
As mentioned in the comments, the only way to answer the question mathematically is by settling on what kind of objects we want to consider, and, afterwards, what the dimension of a given object means.
Now it turns out that mathematicians are generally interested in a plethora of different "objects", many of which have associated notions of dimension. The Wikipedia article on "dimension" gives a bunch of different examples. For the objects you're considering, what you're asking is indeed so reasonable that we might take it to be the definition of the term "dimension", so that we don't even have to proof anything.
To expand a bit on this, note that all of the objects you consider are examples of topological manifolds, which may roughly be described as follows: every point in a topological manifold has a neighbourhood which may be continuously deformed to look like $\mathbb R^n$ for some $n$. For example, in you consider a point on a sphere and zoom in enough, you'll eventually be seeing a plane (we can make maps of the Earth)! The number $n$ turns out to be uniquely determined and is called the dimension of the manifold.
Now, these neighbourhoods you should think of as describing coordinates close to the point, so that, as you say yourself in your comment above, the dimension is simply the number of real numbers making up the coordinates needed to describe points on the manifold. Thus, if your question had been whether or not it's true that the least number of real numbers you need to describe the object, then this would be true simply by definition of "dimension".
However, you're really asking about parameters and not coordinates, so we should also agree on exactly what we mean by the word "parameters". Normally, these would be numbers that together describe the entire space, rather than just small neighbourhoods of points; again, the Wikipedia article provides some discussion of this. One difference that's clear already from your examples is that different values of the parameters might describe the same points (for the circle, by changing $\theta$ to $\theta + 2\pi$, you end up at the same point), whereas the coordinates we really want to uniquely determine a given point (but then, we will generally need several sets of coordinates to describe the entire object). Nonetheless, as you're essentially saying in the comments, you could use these parameters to describe your coordinate neighbourhoods which on the other means that what you're asking is indeed true, more or less by definition.