ANSWER IN YOU OWN WORDS! Don’t forget to provide references

Yes,

Parent can recognize an intangible asset related to the agreement, because as per FASB ASC 805-20-55 if a right on brand or company is transfered to any other company all rights of the subsidary company will transfer to the parent company in the given case five year agreement to supply goods to a customer of subsidary company will be the parent companies agreement also as per 805-20-55 parent company acquires all the customer relationship contracts including all related intellectual  property since the agreement is five year agreeement and also both the parent and subsidary company thinks that the customer will renew the agreement, as per asset definition if any future cash flows are more than one year we can consider it as asset . parent company can recognize  it as intangible asset.

No,

in the second case since it's a one year term autorenewal contract and it can be cancelable by policy holders we can't consider it as intangible asset since there is an uncertainity about the continuation of the contract so parent company can't consider it as a intangible asset.