A Latin square of order \(n\) possessing a cyclic automorphism of order \(n\) is said to be diagonally cyclic because the entires occur in cyclic order down each of the broken diagonals. The author considers a more general setting by considering Latin squares which possess any cyclic automorphism. In particular, he studies what he calls \(B_b\)-type Latin squares. In this more general setting, diagonally cyclic squares are of \(B_0\)-type. These \(B_b\)-type Latin squares have been studied in other contexts and with various names, including Parker squares. The author uses this Parker terminology for \(B_b\)-type squares in honor of E. T. Parker. The author provides an excellent survey of Parker squares, along with some new results, including some results related to sets of mutually orthogonal Parker squares. He also presents an application to the construction of subsquare-free squares along with a conjecture. The definition of \(B_b\)-type Latin squares and the author's results are however, too detailed and complicated to state here.