I would add that there are different kinds of leave that may be available to parents. Under the Canada Labour Code, for employees of the federally regulated private sector, there is a bereavement leave currently available. It's three days paid leave immediately after the death of a loved one, including infants.
Also, many leave provisions under the code were enhanced as part of budget 2017. When those changes do come into force, employees in the federally regulated sector will have access to five days of bereavement leave, the first three of which will be paid. In addition, employees will have the right to request flexible work arrangements as well as leave for family responsibilities. That's to say that these leaves could be used by parents following the tragic death of their child.
We always encourage EI claimants to reach out to their employers to see what other types of benefits or leave may be available to them, but as Andrew mentioned, there's not just the parental benefits leave. Sickness benefits may also be a possibility, allowing the person to continue to be on leave and recover from that tragic event.