At the risk of annoying people by answering something that has been answered already lots of times, I use a solution which used to be available via http://www.aspjson.com/, but the site seems to be down now.
I still use the code though - it's here http://pastebin.com/qvn2UxXV
This is a very simple example of how to use it with your data, to get at deliveryNo and Content:
<!--#INCLUDE file="aspJSON.asp" -->
<%
Set oJSON = New aspJSON
jsonstring = "{""labels"":[{""shipmentID"":""a2f3vh3"",""deliveryNo"":""1234"",""content"":""test"",""invoice"":null,""responseStatus"":{""code"":200,""message"":""SUCCESS"",""messageDetails"":""All shipments processed and label/invoice generated successfully"",""logMsg"":[]}}],""responseStatus"":{""code"":200,""message"":""SUCCESS"",""messageDetails"":""All shipments processed successfully"",""logMsg"":[]}}"
Set oJSON = New aspJSON
oJSON.loadJSON(jsonstring)
For Each result In oJSON.data("labels")
Set this = oJSON.data("labels").item(data)
var_deliveryNo = this.item("deliveryNo")
var_content = this.item("content")
response.write "deliveryNo: " & var_deliveryNo & "<br />"
response.write "content: " & var_content & "<br />"
Next
%>