As long as you just look at a teacup, or a graph of a function, I fully agree with you.
I would say, yes, you do it, and time is the parametrisation you use.
In my opinion, if somebody says that 'y changes as function of the change of x', she is saying 'if you change x from a to b, then y changes according f(a) to f(b)'. But a change of x is an activity of an observer, and observers exist in time.
And concerning your points with MigL: of course change can occur without observers. But observing a change means either a passive observer sees things changing (you are sitting quietly at a veranda, and you see the streets changing from dry to wet (a very common experience in Ireland...)) or you look at an non-changing object, but you let wander your gaze from one place of the object to another, and e.g. see how the colour 'changes' dependent on the place of the object where you look. But the latter also means a change in time. But the change is in the observer, not in the object. The object is static.