Yes, those electoral oversight laws restricted to political parties are constitutional, because they do not violate the "Times, Places, and Manner" clause and have not been preempted by any federal law. Georgia has a similar electoral oversight system. No oversight laws have been challenged to my knowledge.
The U.S. Deptartment of State has a general overview of the administration of elections within each of the 50 states. Most states have a Secretary of State (SoS) or a State Election Board that is in charge of certifying election results and ensuring that elections are fair. Tennessee's State Election Commission lists the duties and composition of the State Election Commission (Note: the Tennessee SoS mentions a 7 member commission, not 5):
- five members elected by joint resolution of the General Assembly, three from the majority party in the legislature and two from the minority party for four year terms.
- appoints County Election Commissioners.
- certifies names of candidates for statewide elections.
- must approve all voting machines/systems before use.
Two other states have organizations called a State Election Commission (SEC), South Carolina and West Virginia. Other states have organizations that have similar duties, but are called State/Local Board of Election Commissioners, State/County Commission/Registrars, etc. The closest system to Tennessee's State Election Commission, is Georgia, where the members of the State Election Board are the Secretary of State, one individual nominated by each of the two political parties, one each by majority vote of the Senate/House.
The constitutionality of Tennessee's State Election Commission being elected by a joint resolution of the General Assembly is a tricky question. The U.S. Dept. of State has an overview of what power it believes Congress has to standardize national elections. The 19th Amendment (women's suffrage), 24th Amendment (poll tax), and 26th Amendment (voting age) set national standards regarding who is allowed to vote. Traditionally, and due to the absence of a constitutional amendment (like those listed above), election responsibilities rested with the states, and each state is given considerable discretion on who can vote, run for office, and the administration of voting within the state. The constitution does lay out some provisions that are delineated to Congress, specifically Article I, §4, cl. 1
The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the Places of chusing [sic] Senators.
It was this "Times, Places and Manner" clause that lead to the recent overturning of Arizona's Proposition 200 by the Supreme Court of the United States (SOCTUS), which contended that it was in violation of the National Voter Registration Act. SCOTUS mentions in its ruling:
The Clause’s substantive scope is broad. “Times, Places, and Manner,” we have written, are “comprehensive words,” which “embrace authority to provide a complete code for congressional elections,” including, as relevant here and as petitioners do not contest, regulations relating to “registration.” Smiley v. Holm, 285 U. S. 355, 366 (1932); see also Roudebush v. Hartke, 405 U. S. 15, 24–25 (1972) (recounts); United States v. Classic, 313 U. S. 299, 320 (1941) (primaries). In practice, the Clause functions
as “a default provision; it invests the States with responsibility for the mechanics of congressional elections, but only so far as Congress declines to preempt state legislative choices.” Foster v. Love, 522 U. S. 67, 69 (1997) (citation omitted).
The election of the State Election Commission does not likely fall within the jurisdiction of the federal government, because no such law (to my knowledge) exists prescribing how states choose to administer electoral oversight. In fact, the existence of so many differences between states election offices, as noted by the US Dept of State, indicates that there probably is not such a law. (Note regarding discrimination of protected classes: The SCOTUS decision (page 14) above states that the 14th Amendment didn't empower Congress to impose the 18-year-old-voting mandate. Neither young age, nor political affiliation are protected classes.) So for it to be unconstitutional, it would need to violate the Tennessee Constitution. Under Article VII. State and County Officers, it says:
Section 1. The qualified voters of each county shall elect for terms of four years a
legislative body, a county executive, a sheriff, a trustee, a register, a county clerk and an assessor of property. [...]
Section 4. The election of officers, and the filling of all vacancies not otherwise directed or provided by this Constitution, shall be made in such manner as the
Legislature shall direct.
Section 4 appears to indicate that the Tennessee Legislature has the authority to determine how any state or county officers are elected.