A nanohertz-frequency stochastic gravitational-wave background can potentially be detected through the precise timing of an array of millisecond pulsars. This background produces low-frequency noise in the pulse arrival times that would have a characteristic spectrum common to all pulsars and a well-defined spatial correlation. Recently the North American Nanohertz Observatory for Gravitational Waves collaboration (NANOGrav) found evidence for the common-spectrum component in their 12.5-year data set. Here we report on a search for the background using the second data release of the Parkes Pulsar Timing Array. If we are forced to choose between the two NANOGrav models $unicode{x2014}$ one with a common-spectrum process and one without $unicode{x2014}$ we find strong support for the common-spectrum process. However, in this paper, we consider the possibility that the analysis suffers from model misspecification. In particular, we present simulated data sets that contain noise with distinctive spectra but show strong evidence for a common-spectrum process under the standard assumptions. The Parkes data show no significant evidence for, or against, the spatially correlated Hellings-Downs signature of the gravitational-wave background. Assuming we did observe the process underlying the spatially uncorrelated component of the background, we infer its amplitude to be $A = 2.2^{+0.4}_{-0.3} times 10^{-15}$ in units of gravitational-wave strain at a frequency of $1, text{yr}^{-1}$. Extensions and combinations of existing and new data sets will improve the prospects of identifying spatial correlations that are necessary to claim a detection of the gravitational-wave background.
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