The elasticity of highly branched polymers, known as bottlebrush polymers, is integral to understanding their physical properties in a wide variety of applications; bottlebrush polymers undergo significant molecular extension in stretched, soft elastomers, in out-of-equilibrium environments during solution processing or in confinement-induced stretching. Unlike linear polymer chains where the molecular origin of this extension is well understood, it remains a challenge to connect molecular bottlebrush architecture to force- extension behavior. In this paper, we present results from Monte Carlo simulations on bottlebrush polymers subjected to a constant pulling force and determine force-extension curves as a function of side-chain length. We show that different bottlebrush architectures exhibit a variety of force-extension behaviors. To understand bottlebrush elasticity, we compare with a parameterized wormlike cylinder implicit side chain model. This demonstrates the emergence of two distinct modes of bottlebrush stretching; at low forces, we demonstrate both linear and non-linear regimes corresponding to stretching the overall cylindrical shape of the molecule, and at high forces, there is specific extension of internal degrees of molecular freedom corresponding to the bottlebrush backbone. This two-regime molecular picture provides insight valuable to the molecular design of bottlebrush materials.