If you need a more generic approach for parsing the connection string (one that doesn't deal with the specifics of one database provider) you can also use
System.Data.Common.DbConnectionStringBuilder
which is a base class for other classes like SqlConnectionStringBuilder etc.
You can create an instance of DbConnectionStringBuilder and in my case I needed to have one configurable connection string that I could get information from -- regardless of the database provider type. A few options if you need this flexibility -- you could create the appropriate ConnectionStringBuilder for your provider as others have suggested -- this would likely be required for most cases where provider-specific properties are needed.
Or if you want to read just a couple generic properties, you could use DbConnectionStringBuilder if you just need the user id and password for example.
This sample should work for ANY connection string that includes user id and password.
DbConnectionStringBuilder db = new DbConnectionStringBuilder();
db.ConnectionString = ConfigurationManager.ConnectionStrings["myConnectionString"].ConnectionString;
var username = db["User Id"].ToString();
var password = db["Password"].ToString();