Peyton DID prioritize parks for about a month. It was a well thought out plan and based on the uber successful Minneapolis model. Minneapolis is considered to have the best Parks and Rec system in the country, with every resident within 6 blocks or less of a park. However, city council opposition to losing control over their park patronage fiefdoms quickly surfaced, and Peyton dropped the idea like a hot potato.
Peyton did court the Police and Fire folks and got their full backing. As memory serves they were less than happy with the results they got. I specifically remember him getting booed at one of the 'Guns and Hoses' [police vs. firefighter] boxing event at the arena during his term.
As far as the courthouse, the original version (as rendered under the Delaney administration for the BJP) had just as big or bigger footprint. It was just a much grander building. It really is a shame that one didn't get built, but construction costs soared between the time it was approved and the time the construction was ready to start. I can't place the blame for that on Peyton too much, and I have never been a fan of him.
The Property tax increases Peyton did were not rolling back the millage to back out the increase in the annual valuation. Not doing so is concerned a tax increase, and Peyton himself called it that when his opponent Matt Carlucci had voted likewise as a member of council. Once elected, Peyton did the exact same thing he criticized Carlucci for doing, and multiple times too. I can't remember if it was twice or three times.
Peyton's predecessor, John Delaney, rollback the millage for the increased valuation and then some, every year for eight years, IIRC. Unfortunately for Peyton, Delaney had the fortune of a growing national economy the whole time. Almost as soon as Peyton took office, the economy turned south in a big way, which put Peyton in a lurch. He ended up adding some new fees to the water bills while in office.