Notice how, unlike in the proposed analyses in the question, the actual text repeats "for." This helps identify the items in the list, making it clear that the correct reading is
The right of citizens of the United States to vote in any primary or other election
- for President or Vice President,
- for electors for President or Vice President, or
- for Senator or Representative in Congress,
shall not be denied or abridged by the United States or any State by reason of failure to pay any poll tax or other tax.
Another way to show this is to reconstitute the two candidate interpretations to see which one makes sense. Take each branch of the "or" and insert it separately into the rest of the sentence:
The right of citizens of the United States to vote in any primary or other election for President or Vice President shall not be denied or abridged by the United States or any State by reason of failure to pay any poll tax or other tax.
The right of citizens of the United States to vote in any primary or other election for electors for President or Vice President shall not be denied or abridged by the United States or any State by reason of failure to pay any poll tax or other tax.
The right of citizens of the United States to vote in any primary or other election for Senator or Representative in Congress shall not be denied or abridged by the United States or any State by reason of failure to pay any poll tax or other tax.
All perfectly fine. What happens if we include the bit about elections only with the first branch?
The right of citizens of the United States to vote in any primary or other election for President or Vice President shall not be denied or abridged by the United States or any State by reason of failure to pay any poll tax or other tax.
The right of citizens of the United States to vote for electors for President or Vice President shall not be denied or abridged by the United States or any State by reason of failure to pay any poll tax or other tax.
The right of citizens of the United States to vote for Senator or Representative in Congress, shall not be denied or abridged by the United States or any State by reason of failure to pay any poll tax or other tax.
If you do this, you're not only taking away the mention of primary elections from the second and third branches, you're taking away all mention of elections, primary and other. What's the purpose of saying "right to vote for electors" and so on but "right to vote in an election for president" and so on?
Besides, "vote for Senator" is imprecise and casual. More formally, you vote for a candidate, not for an office. But it's normal to speak of an election for an office, and normal to speak of voting in such an election.
The first reading honors the principle of parallelism and uses the more formal and precise wording in all three branches of the clause, but even if the second reading were correct it would not have the purported effect. There's no reason to think that "vote for senator or representative in Congress" should not include voting in a primary.
