One-counter automata are obtained by extending classical finite-state automata with a counter whose value can range over non-negative integers and be tested for zero. The updates and tests applicable to the counter can further be made parametric by introducing a set of integer-valued variables. We revisit the parameter synthesis problem for such automata. That is, we ask whether there exists a valuation of the parameters such that all infinite runs of the automaton satisfy some omega-regular property. The problem has been shown to be encodable in a restricted one-alternation fragment of Presburger arithmetic with divisibility. In this work (i) we argue that said fragment of the logic is unfortunately undecidable. Nevertheless, by reduction to a class of partial observation games, (ii) we prove the synthesis problem is decidable. Finally, (iii) we give a polynomial-space algorithm for the problem if parameters can only be used in tests, and not updates, of the counter.