Locations on a hexagonshaped, 72rhombus Parachess board can be identified by “transposed” algebraic notation, with rows indicated by the letters a (White's end) through k (Black's end) and positions within rows numbered from White's left.
Three types of singlespace moves on a Parachess board are

an edgewise step (across a side shared by two rhombuses),


a diagonal step (in a straight line along the main diagonal to the next rhombus), and


an arcwise step (through the long point to a rhombus not connected to the starting space edgewise or diagonally). 
Longer straightline movement (similar to a Rook’s or a Bishop’s move in standard chess) can be defined as a series of singlespace moves:

an Edgepath (a series of edgewise steps with each continuation step crossing the opposite side of the current rhombus),


a Diagonalpath (a series of diagonal steps with each continuation step crossing the opposite corner of the current rhombus), and


a Wavepath (a series of arcwise steps with each continuation step crossing the opposite corner of the current space in the opposite direction of the immediately preceding step).

Twostep leapers  similar to a standard chess Knight  can be developed by combining two different types of onestep moves, with the second step “continuing away” from the starting space: if the first step is edgewise the second step would continue across the opposite edge or the acute angle corner adjacent to the opposite edge (as appropriate), while if the first step is diagonal or arcwise, the second step would continue through the opposite corner or an edge adjacent to the opposite corner.
A piece analogous to a “Knight”, for example, would shift either one step diagonally followed by one step edgewise or one step edgewise followed by one step diagonally to reach its destination. A different type of twostep leaper is the “Archer”, which moves by shifting either one space arcwise followed by one space diagonally or one space diagonally followed by one space arcwise. From a central space (such as e5, as shown in the diagram below) a Knight or Archer each has eight possible leaps (the maximum number for each piece), always to a rhombus of a different color not linked to its starting space by an Edgepath, Diagonalpath or Wavepath:
A significant collection of unusual chessmen can be developed from the basic line movements and the variations on the twostep leap patterns sketched above, but a relatively simple variant that bears some resemblance to standard chess involves the following chess pieces:

King

Moves one step edgewise, diagonally or arcwise to a space where it is safe from capture. 

Pawn

Always moves forward one step edgewise, diagonally or arcwise, but captures only with an edgewise or diagonal move. Upon reaching the opponent’s end rank, a pawn promotes to any piece except a King.


Rook

Moves along Edgepaths. 

Bishop

Moves along either Diagonalpaths or Wavepaths. 

Sorcerer

Leaps two steps away from its starting space: one diagonal step and one step either edgewise or arcwise. 

Queen

Moves along Edgepaths or Diagonalpaths or Wavepaths. 
The Parachess pawn (with four possible moves and up to two captures from any space not on the board’s edge) is more mobile than the standard chess pawn,. A rough guess at the relative values of the pieces (on a scale where pawns = 1) would be Rook = 3, Bishop = 3, Sorcerer (combination of Knight + Archer) = 3.5 (i.e., between 3 and 4) and Queen = 6. The initial placement of the chessman is illustrated below.
In this variant castling is not permitted. A player wins by either checkmating the opposing King or by stalemating the opponent.
A “Zillions of Games” program for this variant (“Parachess  QueenSRB Variant”) also includes a number of other variants involving different sets of Parachess pieces. You can download it here: parachess.zip.