/ AbstractPulsar timing arrays aim to detect nanohertz-frequency gravitational waves (GWs). A background of GWs modulates pulsar arrival times and manifests as a stochastic process, common to all pulsars, with a signature spatial correlation. Here we describe a search for an isotropic stochastic gravitational-wave background (GWB) using observations of 30 millisecond pulsars from the third data release of the Parkes Pulsar Timing Array (PPTA), which spans 18 years. Using current Bayesian inference techniques we recover and characterize a common-spectrum noise process. Represented as a strain spectrum $h_c = A(f/1 {\rm yr}^{-1})^α$, we measure $A=3.1^{+1.3}_{-0.9} \times 10^{-15}$ and $α=-0.45 \pm 0.20$ respectively (median and 68% credible interval). For a spectral index of $α=-2/3$, corresponding to an isotropic background of GWs radiated by inspiraling supermassive black hole binaries, we recover an amplitude of $A=2.04^{+0.25}_{-0.22} \times 10^{-15}$. However, we demonstrate that the apparent signal strength is time-dependent, as the first half of our data set can be used to place an upper limit on $A$ that is in tension with the inferred common-spectrum amplitude using the complete data set. We search for spatial correlations in the observations by hierarchically analyzing individual pulsar pairs, which also allows for significance validation through randomizing pulsar positions on the sky. For a process with $α=-2/3$, we measure spatial correlations consistent with a GWB, with an estimated false-alarm probability of $p \lesssim 0.02$ (approx. $2σ$). The long timing baselines of the PPTA and the access to southern pulsars will continue to play an important role in the International Pulsar Timing Array.