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I am teaching myself more about mathematical logic. I do not know if parenthesis get included in Godel numbering. In the Translated Original Presburger Arithmetic Paper he makes the claim that "the meaningful statements are all built with:" (p. 8 of 21) and doesn't include parenthesis. Yet, when I looked up the Godel numbering for Peano axioms on Wikipedia, they include parenthesis. And in the original paper he calls the statement $\exists\alpha(\alpha+1=0)$ meaningful (p. 10 of 21). Is there a way to write every formula in Presburger arithmetic without parenthesis?

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    From the paper you cited, page 4: "Presburger also uses Lukasiewicz’s parenthesis-free notation with “A” and “K” as prefix, binary operators for “or” and “and” respectively." – 2017-01-30
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    Anyway, the details of the specific notation and encoding being used are not particularly important, since nobody ever actually uses Godel numbering anyway. The important thing is that there is some such encoding. – 2017-01-30
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    I will be using it to prove something for a project. I just want to use the least amount of symbols possible. – 2017-01-30

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You can use prefix notation, e.g. look up 'Polish notation'

E.g. your claim would be something like

$\exists \alpha = + \alpha 1 0$