There are two issues.
First, can the HOA deny access to its facilities and common areas for an unpaid HOA assessment. There might be exceptions in particular jurisdictions, but the general rule would be that it is not unlawful to deny someone who isn't current on their dues use of facilities and common areas.
Second, do you have an unpaid HOA bill?
This is obviously a factual question. The HOA takes the position that you do and is acting accordingly. You could sue the HOA for a determination that your bill is paid in full, or could try to work with the HOA treasurer to determine why the amount that you believe is owed differs from the amount that they believe is owed. I've seen cases where the HOA failed to credit payments that were made or miscalculated the amount due, and I've seen cases where, for example, the homeowner has paid the principle, but not additional amounts owed for late fees, interest and attorneys' fees incurred in collecting the balance that are also owed.