布小熊

I think he shouldn't quit and your husband needs to submit his parental leave request soon.  I don't think anyone can take "parental leave" after he quit or being fired. But one can take EI if he's been laid off.
Request for Leave

The Act says that a request for pregnancy or parental leave must be made in writing at least four weeks before the proposed start date.

More info here:
http://www.labour.gov.bc.ca/esb/facshts/leave.htm