Neither topic is covered as a Common Core standard as such.
Somewhere in middle school, students learn that a circle cannot be a graph of a function because it fails what we call the Vertical Line Test. (I could look up the exact standard and grade level if you're curious about it.) In Geometry (tenth or eleventh grade typically, but it can vary by ...
I am in a US high school.
The implicit equations look like "conics", and are part of the junior (3rd year) class typically called Trigonometry with Algebra. Parametrics are part of the course labeled pre-calculus, which comprises either juniors who will take calculus as seniors, or as seniors in preparation of college calculus.
The community college where I teach puts an introduction (lines, circles, ellipses, parabolas) in precalculus, and then covers it again in vector calculus. So, this would be first year and again in second year of undergrad.