Setup mail notifications on SQL Server (severe) errors
If the SQL Agent Service is not running I'm not getting notified about errors.Therefore I'm wondering, there must be a better way to do what I want?


Few options  

1.Monitor from different server  

IF EXISTS(
SELECT 1
FROM MASTER.dbo.sysprocesses
WHERE program_name = N'SQLAgent - Generic Refresher')


2.Use a Third Party Monitoring service like SQLSentry  

Also i don't see a reason why you need to Parse SQL error log .SQLServer alerts can be used to catch most of the errors/Performance conditions 

below article has more details on how to set up alerting for different scenarios

https://www.simple-talk.com/sql/database-administration/sql-server-alerts-soup-to-nuts/  

And this article also shows  ,few more ways to monitor  using audits