child born of older parents
Refers to a child whose parents are relatively old at the time of birth. The term itself is neutral, but context may carry a nuance of being doted on or having a different upbringing.
He was a child born of older parents; his parents were already in their fifties.
child raised by grandparents; child spoiled by doting grandparents
Often implies that the child is pampered or indulged by grandparents, sometimes leading to a spoiled personality. The term can carry a slightly negative or affectionate tone depending on context.
Spoiled by his grandmother, he became a complete grandparent-raised child.
It's said that children raised by grandparents tend to become selfish.
甘やかす is a verb meaning 'to spoil' or 'to pamper'. 年寄りっ子 often implies such treatment from grandparents.
Compound of 年寄り (toshiyori, 'old person') + っ子 (-kko, diminutive suffix for a child). The suffix っ子 adds a familiar or affectionate tone, often used in words describing a child with a particular characteristic or upbringing.