We consider an initial-boundary value problem for a fully nonlinear coupled parabolic system with nonlinear boundary conditions modelling hygro-thermal behavior of concrete at high temperatures. We prove a global existence of a weak solution to this system on an arbitrary time interval. The main result is proved by an approximation procedure. This consists in proving the existence of solutions to mollified problems using the Leray-Schauder theorem, for which a priori estimates are obtained. The limit then provides a weak solution for the original problem. A practical example illustrates a performance of the model for a problem of a concrete segment exposed to transient heating according to three different fire scenarios. Here, the focus is on the short-term pore pressure build up, which can lead to explosive spalling of concrete and catastrophic failures of concrete structures.