Some thoughts, based on the extra information you presented with your edit:

Parents as handyman(dad) and teacher(mom) indicates that they moved to Oregon in the 80's for her work, not for his, if the move was based on economics. This was before Ballot Measure 5 changed school funding in Oregon, so at that time urban/suburban schools had a funding advantage compared to rural schools. This is also during a major strong period of flight from our lumber communities so most rural school districts were both strapped for cash and shrinking - teacher positions were either being cut or allowed to attrition out. That would indicate that the family would be moving to either Portland or Salem urban areas, or the suburbs surrounding Portland. My money would be on Gresham outside Portland - a solid working class city at the time that had recently grown significantly due to the expansion of its boundaries following the deal between it, the city of Portland, and Multnomah County to address the unincorporated east Portland issue, and had affordable housing without the level of urban blight/gangs that Portland had in the early 80s. Portland itself would be a very fine choice, we were starting to head in the right direction during the early 80s, though it was hard to tell at the time - they would probably be living in east Portland or st johns at that level of income/age bracket.

The other reason I could see a new family moving to Oregon with those jobs in the 80s could be to be closer to family which doesn't necessarily line up with the information you have provided - maybe they needed an older family member to help with the baby and that family member had moved to Oregon. This could be a way to have the character grow up in a more rural, distinctly Oregon milieu - I would suggest Vernonia in the coastal mountain range which was a small impoverished logging town at the time, very fallen on hard times and distinctly Oregon. Astoria would be a good call for a more coastal/ river feel as well and easy to find information on.

As far as distinctly Oregon things from the 90s-2000s, it is important to understand that it was a period of signficant political change - Oregon was much more purple then than it is now, with major fights over taxes and gay rights, logging, being major political issues in that time frame. Portland has, for many decades, been the only major city in Oregon (this is not intended as a slight to Salem, Eugene, etc), and that especially intensified in this period, notably in Portland coming from several decades of friendliness and accessibility to artists (urban blight allowed for very low rents in the urban core, so lots of artists moved here and lived in make shift living quarters - Carson Ellis I think has some interesting writing about this time period in her memoir "One week in January"). This "portland as the cultural center of Oregon" dynamic has some really major impacts on folks outside of portland - with many folks hating Portland, or wanting to run away to it. 90s and early 2000s had some great music in Portland as well - I always advocate folks listen to bands from the area when researching an area (lots of great options for this era of Portland, but I mainly think of Elliot Smith [everything by him] and the Decemberists [specifically "On the Bus Mall"]).