Background/Objectives: Understanding how dual-tasking and Parkinson’s disease medication affect gait and balance regularity can provide valuable insights to patients, caregivers, and clinicians regarding frailty and fall risk. However, dual-task gait and balance studies in PD most often only employ linear measures to describe movement regularity. Some have used nonlinear techniques to analyze PD performances, but only in the on-medication state. Thus, it is unclear how the nonlinear aspects of gait or standing balance are affected by PD medication. This study aimed to assess how dopaminergic medication influenced the regularity and determinism of joint angle and anterior–posterior (AP) and medial–lateral (ML) center of pressure (COP) path time-series data while single- and dual-tasking in PD. Methods: Sixteen subjects with PD completed single- and dual-task gait and standing balance trials for 3 min off and on dopaminergic medication. Sample entropy and percent determinism were calculated for bilateral hip, knee, and shoulder joints, and the AP and ML COP path. Results: There were no relevant medication X task interactions for either the joint angles series or the balance series. Instead, the results supported independent effects of medication, dual-tasking, or standing with eyes closed. Balance task difficulty (i.e., eyes open vs. eyes closed) was detected by the nonlinear analyses, but the nonlinear measures yielded opposing results such that standing with eyes closed simultaneously yielded less regular and more deterministic signals. Conclusions: When juxtaposed with previous findings, these results suggest that medication-induced functional improvements in people with PD might be accompanied by a shift from lesser to greater signal consistency, and the effects of dual-tasking and standing with eyes closed were mixed. Future studies would benefit from including both linear and nonlinear measures to better describe gait and balance performance and signal complexity in people with PD.