We investigate in detail a family of quintessential inflation models in the context of R+R^2 Palatini modified gravity. We find that successful inflation and quintessence are obtained with an inflaton scalar potential that is approximately quadratic in inflation and inverse quartic in quintessence. We show that corrections for the kination period due to Palatini modified gravity are subdominant, while the setup does not challenge constraints on modified gravity from Solar System observations and microscopic experiments, in contrast to the metric case. We obtain concrete predictions regarding primordial tensors to be probed in the near future.