Difficult (if not impossible) to tell from that information.
So, fix it yourself! The first thing to do is to find out exactly what string you are using: so change your code slightly:
string sip = serverIP.Text;
string strCon = "network library = dbmssocn ; Data Source=" + sip + " ;Initial Catalog= Parking_Database ;Integrated Security=True";
SqlConnection connection = new SqlConnection(strCon);
connection.Open();
either use the debugger to examine the content of the "strCon" variable immediately before the new SqlConnection is created, or log it to an output where you can read it later - a log file, or teh console is fine.
Then, try setting up a connection in VS with the Server Explorer pane:
1) Open Server Explorer.
2) Right click "Data connections" and select "Add connection"
3) In the dialog that follows, select your DataSource, and database, specify the security info, and press the "Test connection" button.
4) When the connection works, press "OK"
5) Highlight your database in the Server Explorer pane, and look at the Properties pane. A working example of the connection string will be shown, which you can copy and paste into your app or config file.
Compare that to the string you are using, and the problem may become clearer.