  Gravitational waves are predicted by Einstein's theory of general relativity
as well as other theories of gravity. The rotational stability of the fastest
pulsars means that timing of an array of these objects can be used to detect
and investigate gravitational waves. Simultaneously, however, pulsar timing is
used to estimate spin period, period derivative, astrometric, and binary
parameters. Here we calculate the effects that a stochastic background of
gravitational waves has on pulsar timing parameters through the use of
simulations and data from the millisecond pulsars PSR J0437--4715 and PSR
J1713+0747. We show that the reported timing uncertainties become
underestimated with increasing background amplitude by up to a factor of
$\sim10$ for a stochastic gravitational-wave background amplitude of $A=5\times
10^{-15}$, where $A$ is the amplitude of the characteristic strain spectrum at
one-year gravitational wave periods. We find evidence for prominent
low-frequency spectral leakage in simulated data sets including a stochastic
gravitational-wave background. We use these simulations along with independent
Very Long Baseline Interferometry (VLBI) measurements of parallax to set a
2--sigma upper limit of $A\le9.1\times 10^{-14}$. We find that different
supermassive black hole assembly scenarios do not have a significant effect on
the calculated upper limits. We also test the effects that ultralow--frequency
(10$^{-12}$--10$^{-9}$ Hz) gravitational waves have on binary pulsar parameter
measurements and find that the corruption of these parameters is less than
those due to $10^{-9}$--$10^{-7}$ Hz gravitational waves.
