Right, but there are a lot of federal agencies that adjudicate and potentially give out harsh fines without actually being a court. If you are in that situation you do not have the right to a jury trial, at least until this case was decided.

That's not true for all civil cases, though. The Seventh Amendment says, "In suits at common law, where the value in controversy shall exceed twenty dollars, the right of trial by jury shall be preserved, ..." As has been described elsewhere here, this was interpreted to provide a jury right in suits at law but not in equity. You can also waive your jury right by agreeing not to invoke it, say in a contract.

Can I for my speeding ticket or parking ticket?

Unless you signed an agreement that states otherwise... Most employee and contractor and user agreements fall into this category.

Mandatory arbitration clauses seem to hold up in court fairly regularly.