There is accumulating evidence that numerical information influences the way in which we perform bodily movements. Specifically, the idea that our cognitive representations of numbers and space interact is supported by systematic associations of space with both number magnitude (SNARC effect) and number parity (MARC effect). However, whether this influence is bound to the left or right side of space or to the hand with which we perform the movement remains debated. One novel and interesting way to disentangle these factors is to use movement responses in which hand and movement direction can be dissociated. In the present study, participants moved a central object to the left or right side on a touchscreen with their index fingers as response to a parity judgment and magnitude classification task. We observed significant SNARC effects in both tasks. Number magnitude and response direction interacted, but magnitude and response hand did not. This indicated that the SNARC effect can be independent of the responding hand. Importantly, however, a MARC effect was observed not only in an interaction between response direction and parity, but also in an interaction between response hand and parity, suggesting that response hand plays a role in the interaction between physical space and parity. Additionally, number magnitude influenced the amplitude of participants’ response movements, with larger numbers eliciting longer movements. These results indicate that space, magnitude and parity interact on different levels that can be unraveled in a paradigm utilizing continuous movements such as swiping.