This is a style issue, and there is no absolute—or even especially firm—rule governing how to handle it. If you generally follow the style guidelines in The Chicago Manual of Style, you may be interested in this item from the fifteenth edition (2003):
6.103 With other punctuation. ... Parentheses should rarely appear back to back. Different kinds of material may, if necessary, be enclosed in a single set of parentheses, usually separated by a semicolon. ...
[Relevant example:] (see Smith and Hawkins 1990; t-tests are used here)
Chicago would seem to favor running your side-by-side parentheses as "(PACC; Attachment 1)." In this case, however, I don't think that's a terribly appealing combination as rendered. If I were taking the Chicago approach, I would be inclined to separate the two parentheticals with a transition verb: "(PACC; see Attachment 1)."
But I don't see anything fundamentally unacceptable about running parentheticals back to back, especially if your article has multiple other parentheticals for acronyms/intialisms and for attachments. And even Chicago hedges its guideline with the wording "Parentheses should rarely" instead of "Parentheses should not."
If I really wanted to avoid back-to-back parentheticals, I would consider reworking the sentence to avoid the juxtaposition. For example, if "Attachment 1" is the list of questions from the other person's letter, you might write this:
Thank you for your letter of October 27, 2015, in which you outlined questions (Attachment 1) regarding the development of the new Pima Animal Care Center (PACC).
Not knowing the contents of Attachment 1, I can't say where it should go and how it should be introduced, but it appears to me to be the simplest variable to adjust in order to avoid back-to-back sets of parentheses, if that's what you want to do.