We report on the nonequilibrium quasiparticle dynamics in BaFe$_2$As$_2$ on both the hole doped (Ba$_{1-x}$K$_x$Fe$_2$As$_2$) and electron doped (BaFe$_{2-y}$Co$_y$As$_2$) sides of the phase diagram using ultrafast pump-probe spectroscopy. Below $T_c$, measurements conducted at low photoinjected quasiparticle densities in the optimally and overdoped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ samples reveal two distinct relaxation processes: a fast component whose decay rate increases linearly with excitation density and a slow component with an excitation density independent decay rate. We argue that these two processes reflect the recombination of quasiparticles in the two hole bands through intraband and interband processes. We also find that the thermal recombination rate of quasiparticles increases quadratically with temperature in these samples. The temperature and excitation density dependence of the decays indicates fully gapped hole bands and nodal or very anisotropic electron bands. At higher excitation densities and lower hole dopings, the dependence of the dynamics on quasiparticle density disappears as the data are more readily understood in terms of a model which accounts for the quasiequilibrium temperature attained by the sample. In the BaFe$_{2-y}$Co$_y$As$_2$ samples, dependence of the recombination rate on quasiparticle density at low dopings (i.e., $y=0.12$) is suppressed upon submergence of the inner hole band and quasiparticle relaxation occurs in a slow, density independent manner.
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