My parents told me to respect the elderly.

Literal

Parents [topic-は] me [to-に] elderly [object-を] respect [manner-ように] said.

~ように言う (to tell someone to do X) uses ように to frame a directive. 敬う (to respect/revere) is a somewhat literary verb. 老人 (elderly person) is a direct, neutral term in Japanese — not impolite the way 'old person' can sound in English.