Dopamine (DA) and ascorbic acid (AA) are two important biomarkers with similar oxidation potentials. To facilitate their simultaneous electrochemical detection, a new voltammetric sensor was developed by modifying a screen-printed carbon electrode (SPCE) with a newly synthesized block copolymer (poly(DMAEMA-b-styrene), PDbS) as a dispersant for reduced graphene oxide (rGO). The prepared PDbS–rGO and the modified SPCE were characterized using a range of physical and electrochemical techniques including Raman spectroscopy, scanning electron microscopy, transmission electron microscopy, cyclic voltammetry, electrochemical impedance spectroscopy, and linear sweep voltammetry. Compared to the bare SPCE, the PDbS–rGO-modified SPCE (PDbS–rGO/SPCE) showed better sensitivity and peak-to-peak separation for DA and AA in mixed solutions. Under the optimum conditions, the dynamic linear ranges for DA and AA were 0.1–300 and 10–1100 µM, and the detection limits were 0.134 and 0.88 µM (S/N = 3), respectively. Furthermore, PDbS–rGO/SPCE exhibited considerably enhanced anti-interference capability, high reproducibility, and storage stability for four weeks. The practical potential of the PDbS–rGO/SPCE sensor for measuring DA and AA was demonstrated using ex vivo brain tissues from a Parkinson’s disease mouse model and the control.