Computing parametric sensitivities for oscillators has a now well-understood subtlety associated with the indeterminacy of phase. A less universal, but still vexing, subtlety arises when an oscillator is described by a system of differential equations with "hidden" conservation constraints (HCC{\textquoteright}s); defined as weighted sums of state variables that are time-invariant. If there are HCC{\textquoteright}s, as is commonly the case for models of biochemical oscillators but rarely the case for practical circuit oscillators, the now-standard approach to computing parametric sensitivities can yield incorrect results. In addition, the monodromy matrix (the matrix of state sensitivities over one oscillation period), is often defective in a way that interferes with the usual approach to computing oscillator phase noise. In this paper we analyze the HCC case, and show that by augmenting the standard sensitivity approach with explicit HCC{\textquoteright}s, one can recover the correct parametric sensitivities. In addition, we prove that there is a typically satisfied condition that guarantees that a system with HCCs will have a defective monodromy matrix. A deliberately "flawed" ring oscillator circuit and a cyanobacterial circadian clock biochemical oscillator are used to demonstrate the parametric sensitivity problem and its resolution, and to show the issue of the defective monodromy matrix.