Local political parties get all of their funds from donations, although the mechanics of primary elections (voting machines, space rental, ballot counting staff) are often publicly funded.
In states that have caucus systems, passing the hat at election year caucuses to attendees is often the primary source of revenue (as well as one of the biggest expenses in the local party budget). Other important sources of revenue are an annual dinner with famous keynote speakers, collection of dues from sub-organizations of the local parties like precinct or house district organizations, contributions at summer picnics, a variety of fund raisers, and typically some sort of "club" of regular contributors (e.g. one local party I was involved with had a "century club" for people who gave at least $100 per year to the local party).
Like most fundraising ventures for non-profits, perhaps 20% of donors account for 80% of donations.
Sometimes a local party with have related "committees" such as a "Hardin County County Officials Committee" that operate in a manner similar to committees for Senatorial and Congressional campaigns.
Financial assistance from state parties can happen, but tends to be uncommon and targeted for a specific purpose (e.g. getting all local parties onto a state election database).
Typically, a local political party organization will have far less money at its disposal than the candidates who run in that local party's jurisdiction combined and it isn't uncommon for candidates who are eliminated in primary elections or lose races with money in the bank to turn that money over to the local political party organization.
It also isn't uncommon for a lot of the needs of a local political party to be provided "in kind" rather than in cash, and some local political parties have "wish lists" of particular items that they need on their web sites.
(This answer is based mostly upon my experience serving as treasurer for a county level political party.)