I'll take your more specific question.

The sources I've read about the creation of parks in cities (specifically Central Park, which I'll look through my house to find) shows that during the 19th Century there was some municipally centered demands to create jobs, but this was not to create jobs in and of itself, but for the purpose of creating, as stated, public parks.

The New Deal was really the time when the idea of "job creation" came into its own. Before this time the dominant school of economic thought was the Classical School (Smith, Ricardo, etc) which denied the existence of involuntary unemployment. Basically most people believed that unemployment could only result from people refusing to work below certain wages, not that in economic crises people could be out of work and unable to find it again.

However, by 1933, that was proving to be untrue, John Maynard Keynes also helped by basically disproving it completely in "The General Theory." At this point the Roosevelt Administration were looking for some way to alleviate the crisis, and eventually started the programs composing the New Deal in 1933. A whole portion of the basic program was Relief: meaning job creation through public works to decrease unemployment. After that Keynesian thought dominated economic thought, especially the idea of job creation.

I think your question could be better phrased by reversing it: when did it become expected that governments should not attempt to intervene in economics? I'd argue that it's only been in the recent liberal era that governments were not expected to intervene.

As far as I know, as far back as ancient Athens there was some system of welfare. Citizens who attended the assemblies for a full day were remunerated, to ensure that even poor citizens could participate in the democratic process. Additionally, (many) public offices were doled out by lottery to the citizen body. While it did not give everyone a stable or long-term job, it at least provided some citizens with a livelihood.

Let's also not forget that in Feudal times, lords were expected to provide care and residence for their serfs. Of course, it was not the "state" per se that took care of the serfs, but certainly the manor system in Europe can be seen as a micro-governmental system, especially given the frailty of feudal states. While serfdom was by no means an ideal labour system, it of course offered a stability that disappeared with the rise of the global capitalist system.

Could you specify where? Like, Europe? The United States? In Spain, I know of documents as far back as the eighteenth century that use liberal rationalization for governments directly intervening in the economy, and many of those documents also considered a sort of social contract of labor, calling for the government to ensure that people are able to find work and demanding that those who can work be forced to (essentially the type of logic that was the root of subsequent anti-vagrancy laws throughout the nineteenth century in the British and Spanish Americas).