Regarding judges and inspectors:
RCW 29A.44.410 states that each precinct with polling places (or precincts temporarily consolidated as a single precinct for that primary or election) shall have one judge and two inspectors. These people shall be chosen from a list provided by the county chair of the state's major parties as outlined in
RCW 29A.44.430. The party from which the inspector and judges are selected is determined by "that political party which polled the highest number of votes in the county for its candidate for president at the last preceding presidential election and one judge from that political party polling the next highest number of votes in the county for its candidate for president at the same election." Election rules (not statutes) allow major parties to require persons operating as representatives of the party to sign a loyalty oath before being allowed to participate. "Operating as a representative of the party" includes functions such as caucus participant and election as a PCO (oddly enough, candidates for other elected offices are exempt. I think this is because they are considered as representatives of the people, not the parties.) All parties have historically enforced this requirement.
Regarding clerks:
RCW 29A.44.420 states that any additional clerks shall "represent a major political party. The political party representation of a single set of precinct election officers shall, whenever possible, be equal but, in any event, no single political party shall be represented by more than a majority of one at each polling place." The "operating as a representatives of the party" loyalty oath mentioned above also applies to clerks.
Washington's laws regarding elections are found in the Revised Code of Washington,
Title 29A. If you are curious about the definition of "major party" and the corresponding "minor party", those are given in
RCW 29A.04.086 and
RCW 29A.04.097, respectively.