The Box–Behnken experimental design was used to investigate the effect of subcritical water extraction parameters such as temperature, process duration, and extractor shape on the extract composition and antioxidant activity of Norway maple (Acer platanoides L.) bark extracts. Spectrophotometric (UV-Vis) techniques were employed to evaluate the total polyphenols (TPC) and flavonoids (TFC). The DPPH radical scavenging method was used to evaluate the antioxidant activity of the extracts. The yield of the process was evaluated through the utilization of response surface methodology (RSM). The total polyphenol and flavonoid contents, together with antioxidant activity, are highly dependent on water temperature. The influence of changes in the process duration and the shape of the pressure cell was not observed. A temperature increase from 110 °C to 170 °C caused a 8.9-fold increase in the polyphenol content, 7.2-fold increase in the flavonoid content, and 12.6-fold increase in the antioxidant activity. The highest values for polyphenols, flavonoids, and antioxidant activity occurred at a temperature of 170 °C, which is the upper limit of the temperature variability range for these studies. This study demonstrates the importance of the appropriate selection of extraction parameters in order to obtain the desired chemical composition of the extract.