I want to have 5-20 different sites to give that support, and want it centralized to one support system, that will be installed in one place and in one database.

So for each site, I want one "department" or "company" with one specific inbound mail adress and one specific outbound address.

Email Parsing is also a requirement.

That's all.

Can you advice which support ticket helpdesk can provide me that with a reasonable cost?

Ohhh .... I would also prefer the outbound addresses to have different SMTP server.

I'm pretty sure you can do this with Kayako. Not sure about the different SMTP servers though.