The superconducting penetration depth, $lambda(T)$, has been measured in RFeAsO$_{0.9}$F$_{0.1}$ (R=La,Nd) single crystals (R-1111). In Nd-1111, we find an upturn in $lambda(T)$ upon cooling and attribute it to the paramagnetism of the Nd ions, similar to the case of the electron-doped cuprate Nd-Ce-Cu-O. After the correction for paramagnetism, the London penetration depth variation is found to follow a power-law behavior, $Delta lambda_L(T)propto T^{2}$ at low temperatures. The same $T^2$ variation of $lambda(T)$ was found in non-magnetic La-1111 crystals. Analysis of the superfluid density and of penetration depth anisotropy over the full temperature range is consistent with two-gap superconductivity. Based on this and on our previous work, we conclude that both the RFeAsO (1111) and BaFe$_2$As$_2$ (122) families of pnictide superconductors exhibit unconventional two-gap superconductivity.