You could make a buffer along your line, then convert this buffer to polyline and finally use snap tool to snap points on this line. Python code should looks like this:
import arcpy
# Set workspace
arcpy.env.workspace = r"path_to_your_workspace"
# Set variables
line_feature = "your_line_feautre.shp"
point_feaure = "your_point_feature.shp"
distance_from_line = "10 Meters" # distance for buffer
buffer_feaure = "in_memory" + "\\" + "buffer" # temporary feature
line_from_buffer = "in_memory" + "\\" + "line" # temporary feauture
# Make buffer along line
arcpy.Buffer_analysis(line_feature, buffer_feaure, distance_from_line)
# Convert buffer to polyline
arcpy.PolygonToLine_management(buffer_feaure, line_from_buffer)
# Snap points to line
distance_for_snap = "20 Meters" # distance for include points to snap
arcpy.Snap_edit(point_feaure, [[line_from_buffer, "EDGE", distance_for_snap]])
# Delete
arcpy.Delete_management("in_memory")
You could see the help for tools: Buffer, Polygon to Line and Snap tool.