When several limited power devices are available, one of the most efficient ways to take advantage of these resources, is to run in parallel several neural sub-networks and to fuse the result at the end of processing. However, such a combination of sub-networks must be trained specifically for each configuration of devices (characterized by number of devices and their capacities) which may vary over different model deployments and even within the same deployment. In this work we introduce parallelly distributable slimmable (ParaDiS) neural networks that are splittable in parallel among various device configurations without retraining. While inspired by slimmable networks, allowing instant adaptation to resources on just one device, ParaDiS networks consist of several multi-device distributable configurations or switches that strongly share the parameters between them. We evaluate ParaDiS framework on MobileNet v1 and ResNet-50 architectures on ImageNet classification task and on WDSR architecture for image super-resolution task. We show that ParaDiS switches achieve similar or better accuracy than the individual models, i.e., distributed models of the same structure trained individually (without parameter sharing). Moreover, we show that, as compared to universally slimmable networks that are not distributable, the accuracy of distributable ParaDiS switches either does not drop at all or drops by a maximum of 1 % only in the worst cases. Finally, once distributed over several devices, ParaDiS outperforms greatly (universally) slimmable models.