Personally, I would convert the wetland into a raster (Feature to Raster, information on this can be found here), and then perform a Zonal Statistics analysis (more info here) to determine how much of the raster overlaps with each polygon. You can then join the the statistics table back to the polygon shapefile (the parcels) to see what the pixel count is.
To put it as as simply as possible, if you convert the wetland polygons into a raster file, it will represent the wetlands as a collection of individual pixels in a grid, rather than polygons. The zonal statistics tool can then count how many pixels of the value "wetland", how many pixels of the value "not wetland", and how many pixels total are located within each parcel polygon. From these numbers, you could calculated the percentage of wetland within each parcel polygon.
Please let me know if there's anything else you need me to expand upon!