To get a row count of each combination you can use Summary in the Tables menu. Enter the three year-columns as grouping variables. That will give you a table with a column "N Rows" that represent the number of rows for each 1-0-0, 1-1-0 etc.
The parallel plot is a way to visualize data across rows. But otherwise JMP provides more options with data arranged in columns. The Stack command (Tables menu) is an easy way to stack a horizontal data set that automatically creates a grouping columns based on the names of stacked columns.