Not necessarily. Hopefully this has been decided by your parents before it is necessary . That is what a will is for. Parents usually choose the child who they feel should be in charge and that may or may not be the oldest. Being firstborn doesn't rest that responsibility on one just by virtue of birth order...Unless you are an only child.
MPA/Executor. Anyone over 50 or in questionable health should prepare documents to save their family any confusion over difficult final decisions. A medical power of attorney is a legal form that spells out who will make medical decisions if a person becomes incapacitated.The will should appoint an executor to carry out its provisions including final burial plans if not made ahead of time.