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 district), students learn to calculate the standard and general equations for a circle based on its radius and the coordinate of its center and working backwards from there, but we don't describe it as an implicit function. There are no common standards for pre-calculus at the moment, but people generally cover conics to some level of detail. Implicit differentiation is covered in the AP Calculus standards (both AB and BC), but IME we don't miss the lack of prior knowledge.
When I was in high school 35 years ago, we did cover parametric equations in what was essentially pre-calculus, and I believe differentiating parametric equations came along with studying the Chain Rule in Calc AB. I can easily imagine that some college courses would still cover it, but it is no longer in the AP Calculus standards.