You're more right than you might have expected, but not because it's a fallacy or misleading. You noticed something important in how it all works: time is a dimension, but it doesn't act like "up" or "forwards".
This doesn't make it less of a dimension or a hindrance to understanding, it's an observation that leads to: there are different types of dimensions.
Typically called time like and space like, they can also be thought of as "one directional" and "two directional", although a physicist somewhere is (correctly) coughing politely and glaring at some of the shit photons get up to at the thought of one directional time.

You're thinking of time as a parameter, which is how it is in classical mechanics. It's a different category of thing, but it technically makes the system 4d.
When you start looking at how light moves and relativity you find that you actually need time to act much more like another direction because it no longer defined an order or sequence, and you get stuff like "time slows down when move faster in space because acceleration shifts your movement vector in space time".

It's even simpler in math, because a dimension is simply a number required to specify a point in a space. If you cared to you could use "left" as your parameter and talk about how a thrown ball changes position in time, up, and forward as a function of left.
Then you could do some real math and use that function as a point in some space and talk about how the different components are different dimensional aspects of the infinite dimensional polynomial function space.