It's really an either/or proposition, i.e. either space is expanding or the time experienced by distant objects is dilated, depending on how you view the situation. We choose the former description because it is better.
To expand (excuse the pun!) on what I mean, the measurable result of time dilation is red-shift and indeed distant cosmological objects appear red-shifted and thus "slowed down". It may be tempting to think of faraway objects as being time dilated, however time dilation is not the only cause of red-shift. For example, in special relativity, time dilation is not defined directly from red-shift, but from the relativistic correction to non-relativistic Doppler shift.
In cosmological models we choose coordinates where the red-shift is implicitly and entirely caused by the expansion of space as opposed to time dilation. We could take a different view and try to make the red-shift as entirely due to gravitational time dilation, except the concept of gravitational time dilation doesn't fit that well with cosmological spacetimes and the expansion of space fits much better with the idea of a homogeneous, isotropic Universe.
In summary there is no cosmological time dilation as an analogy to cosmological expansion, at least in the way that we choose to view the situation.