You should just need to add this after your selection.arcpy.RefreshActiveView()
I made a few other changes if you are interested.#import modules and define workspace
import arcpy
import os
import traceback
workspace = arcpy.env.workspace = r"C:/temp"
print workspace
arcpy.env.overwriteOutput = True
#Define map document
mxDoc = arcpy.mapping.MapDocument(r"C:/temp/PSL_Parcel.mxd")
print mxDoc
#List the first dataframe (named Layers) in the map document
df = arcpy.mapping.ListDataFrames(mxDoc, "Layers")[0]
#List first map layer (which is the parcels layer) in dataframe
parcelLayer = arcpy.mapping.ListLayers(mxDoc,"",df)[0]
print parcelLayer
parcelField = "APN_NU_1"
#Create a feature layer to make selections on
selectingLayer = arcpy.MakeFeatureLayer_management(parcelLayer, "parcelLayer_lyr")
row, rows = None, None
#Set up cursor:
rows = arcpy.SearchCursor(selectingLayer)
apnList = []
#Find the field, select each record and get values
for row in rows:
apnList.append(row.getValue(parcelField))
for APN in apnList:
print(APN)
whereClause = "APN_NU_1 = '{0}'".format(APN)
arcpy.management.SelectLayerByAttribute(selectingLayer, "NEW_SELECTION", whereClause)
arcpy.RefreshActiveView()
#Export map with selected record to PDF
pdfLocation = 'C:/temp'
pdfName = os.path.join(pdfLocation, APN + '.pdf')
arcpy.mapping.ExportToPDF(mxDoc, pdfName)
print len(apnList)
del row, rows
#Clean up feature layers from memory and to debug and or rerun script
arcpy.Delete_management(selectingLayer)