Creating a relationship when filling in a Profile
It's not currently possible to create a relationship on a CiviCRM profile form similar to the Employer one when using WordPress or Joomla. 

However, if you create an auto-reference field you can use that to create a link between a participant and another contact. 

You can create the field on all contacts, or a contact subtype that covers all those filling in the event registration for trainingeven, or potentially registrants or just the registrants for that event. My guess is that a registrant auto-reference field is less likely to be convenient or what you want. 

Don't forget that 1) the auto-reference contacts can be limited to a particular group of contacts, and 2) there are security issues in exposing contact data when using contact reference fields on public forms.