Is there an etymological link between the word parabens in English (ex. Soap without parabens) and parabéns (congratulations) in Portuguese?
No, English paraben is from para-hydroxybenzoic acid, and Portuguese parabéns is from para + bem.
For chemical compounds, para comes from Greek παρά (pará) 'besides', and benzoic has its root in Arabic. See Wiktionary.
The two components of the Portuguese words are both from Latin. Here, para is from Latin per + ad. Per is not related to Greek παρά (pará), but to Greek περί (perí).