So here are the juicy details:
Anne was convinced that Catherine Parr possessed a lower rank than herself by marrying Thomas Seymour who was the younger brother of Anne's husband - and therefore far less important. When Anne was asked to carry Catherine's train, the Duchess refused. But the scandal peaked when Anne took the Queen's place at the hearing of matins in the chapel. Allegedly Catherine then grabbed Anne's arm.
In the end Catherine Parr won the rivalry because she referred to the Third Succession Act that clearly stated that Catherine had precedence over any lady at court. It also meant - probably to Anne's horror - that Anne also came after Lady Mary, Lady Elizabeth and Anne of Cleves.
Whether the story is true is difficult to know for sure because the only source (at the moment) is the Spanish Chronicle that is known for stating rather unlikely things.