The Parker instability, a Rayleigh–Taylor-like instability of thermal gas supported against gravity by magnetic fields and cosmic rays, is thought to be dynamically important for galaxy evolution, possibly promoting molecular cloud formation and the galactic dynamo. In previous work, we examined the effect of three different cosmic-ray transport models on the Parker instability: decoupled (γ c = 0), locked to the thermal gas (γ c = 4/3), and coupled to the gas with streaming by self-confinement. We expand on that work here by considering radiative cooling, a smooth gravitational potential, and simulations into the nonlinear regime. We determine that cosmic-ray transport away from compression points, whether by diffusion or streaming, is the largest driver of the instability. Heating due to cosmic-ray streaming is also destabilizing and especially affects the nonlinear regime. While cooling depressurizes the dense gas, streaming cosmic rays heat and inflate the diffuse extraplanar gas, greatly modifying the phase structure of the medium. In 3D, we find that the fastest growth favors short-wavelength modes in the horizontal direction perpendicular to the background magnetic field; this is imprinted on Faraday rotation measure maps that may be used to detect the Parker instability. The modifications to the Parker instability that we observe in this work have large implications for the structure and evolution of galaxies, and they highlight the major role that cosmic rays play in shaping their environments.