Current cosmological tensions motivate investigating extensions to the standard $Lambda$CDM model. Additional model parameters are typically varied one or two at a time, in a series of separate tests. The purpose of this paper is to highlight that information is lost by not also examining the correlations between these additional parameters, which arise when their effects on model predictions are similar, even if the parameters are not varied simultaneously. We show how these correlations can be quantified with simulations and Markov Chain Monte Carlo (MCMC) methods. As an example, we assume that $Lambda$CDM is the true underlying model, and calculate the correlations expected between the phenomenological lensing amplitude parameter, $A_L$, the running of the spectral index, $n_{rm run}$, and the primordial helium mass fraction, $Y_P$, when these parameters are varied one at a time along with the $Lambda$CDM parameters in fits to the $textit{Planck}$ 2015 temperature power spectrum. These correlations are not small, ranging from 0.31 ($A_L-n_{rm run}$) to $-0.93$ ($n_{rm run}-Y_P$). We find that the values of these three parameters from the $textit{Planck}$ data are consistent with $Lambda$CDM expectations within $0.9sigma$ when the correlations are accounted for. This does not explain the 1.8-2.7$sigma$ $textit{Planck}$ preference for $A_L>1$, but provides an additional $Lambda$CDM consistency test. For example, if $A_L>1$ was a symptom of an underlying systematic error or some real but unknown physical effect that also produced spurious correlations with $n_{rm run}$ or $Y_P$ our test might have revealed this. We recommend that future cosmological analyses examine correlations between additional model parameters in addition to investigating them separately, one a time.
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