  A general approach is presented that offers exact analytical solutions for
the time-evolution of quantum spin systems during parametric waveforms of
arbitrary functions of time. The proposed method utilises the \emph{path-sum}
method that relies on the algebraic and combinatorial properties of walks on
graphs. A full mathematical treatment of the proposed formalism is presented,
accompanied by an implementation in \textsc{Matlab}. Using computation of the
spin dynamics of monopartite, bipartite, and tripartite quantum spin systems
under chirped pulses as exemplar parametric waveforms, it is demonstrated that
the proposed method consistently outperforms conventional numerical methods,
including ODE integrators and piecewise-constant propagator approximations.
