Does including a first initial with a last name affect the order of names in a parenthetical citation?

No. When more than one name appears in a parenthetical citation, the order of those names should correspond to the quotations that precede the citation. For example, if a sentence contains two quotations, the first from an author whose last name is Hill and the second from an author whose last name is Barth, then Hill should be listed first in the parenthetical citation. If what precedes the citation is a paraphrase, then the order of names is up to you. If in this case you choose to list the authors in alphabetical order, the names should be ordered by each author’s last name. The author’s first name or initial should not be included in parenthetical citations unless you are citing two authors with the same last name