In parallel-server systems with a single stream of arrivals (a.k.a. load balancing), Join-the-Shortest-Queue (JSQ) is a popular routing algorithm. There is extensive literature studying this system in various asymptotic regimes, but all assume constant parameters (arrival and service rates). We study the JSQ system with Markov-modulated parameters and heterogeneous servers. Our main contributions are: (i) We compute the heavy-traffic distribution of the scaled vector of queue lengths; (ii) We utilize a novel hybrid methodology that combines the Transform Method for queue-lengths analysis and the Poisson equation; (iii) We provide sufficient conditions to ensure state space collapse, showing provable balancing power of JSQ for heterogeneous servers. Unlike other studies involving Markov-modulated queues, these conditions don't depend on the mixing time of the modulating chain and are valid for a countably infinite state space. We numerically demonstrate the strength of our results under moderate traffic intensities and showcase their independence from the corresponding mixing times.