Just commenting to add some context regarding Parallel Plots, vs the more general Line Plots.
In JMP, Parallel Plots require every data point in each line to be in its own column. Your columns would be "January", "February", "March"... Each column would have one row per Year. This is a Wide data format.
If your data has one column called "Month", with entries "January", "February", "March"..., then the plot you need is a Line Plot, created using Graph Builder. This is a Tall data format.
How you overlay the lines from each Year will also depend on how your table is formatted. If you have a single "Month" column, plus an additional "Year" column, then @Thierry_S's approach is the way to go. If instead you have six "Month" columns, one for each Year, then you can just plot all those columns together on the same Line Plot. If you'd like to upload your data table, we could give a more specific answer.
A nice explanation of this topic and how to work with Wide and Tall data formats can be found in the Exploratory Data Analysis (EDA) Tutorial | JMP. Module 2, Section 2.6, "Restructuring Data for Analysis".