That’s a new NBER working paper by Christian Dippel (UCLA), which finds that cities with Democratic mayors are more likely to underfund pensions:

Using data covering a wide range of municipal public-sector pension plans from 1962– 2014, I establish that unfunded pension benefits grow faster under Democratic-party mayors, using a regression discontinuity design (RDD) focusing on narrow mayoral races. Previous evidence shows that parties do not matter for a range of fiscal outcomes in U.S. cities, and suggests this is because Tiebout sorting imposes fiscal discipline. This paper shows that parties do matter for types of fiscal spending where benefits accrue to narrow constituencies and where costs are difficult to observe and understand for tax payers.

Generally the literature shows political parties does not much matter for local government finances, but as the authors here note those studies are typically exploring more visible fiscal outcomes. Here is the discontinuity for the main result (N=1,195):

The paper shows the finding to be robust to a number of subsamples, including plan type (police and fire-fighters), city system (mayor or council managers), and whether the mayor was a challenger or incumbent. The results are most pronounced for plan type, and as the figure shows, the effect is driven by being particularly close to the cut-off. These results imply that it is the result of pork barrel projects to win close elections. With any discontinuity there is concern that the threshold result will be different from the average treatment effect, so we should be a bit hesitant to think that states dominated with one party rule (i.e. no close elections) will have party differences in pension funding.