  We have searched three Parkes multibeam 1.4 GHz surveys for the presence of
fast radio bursts (FRBs) out to a dispersion measure (DM) of 5000 pc cm$^{-3}$.
These surveys originally targeted the Magellanic Clouds (in two cases) and
unidentified gamma-ray sources at mid-Galactic latitudes (in the third case)
for new radio pulsars. In previous processing, none of these surveys were
searched to such a high DM limit. The surveys had a combined total of 719 hr of
Parkes multibeam on-sky time. One known FRB, 010724, was present in our data
and was detected in our analysis but no new FRBs were found. After adding in
the on-sky Parkes time from these three surveys to the on-sky time (7512 hr)
from the five Parkes surveys analysed by Rane et al., all of which have now
been searched to high DM limits, we improve the constraint on the all-sky rate
of FRBs above a fluence level of 3.8 Jy ms at 1.4 GHz to $R = 3.3^{+3.7}_{-2.2}
\times 10^{3}$ events per day per sky (at the 99% confidence level). Future
Parkes surveys that accumulate additional multibeam on-sky time (such as the
ongoing high-resolution Parkes survey of the LMC) can be combined with these
results to further constrain the all-sky FRB rate.
