The process by which a modified stem or root of a parent plant grows offspring that remain attached is called

vegetative reproduction.

The process by which a modified stem or root of a parent plant grows offspring that remain attached is called vegetative propagation. Vegetative propagation is a form of asexual reproduction in plants, where new individuals are produced from vegetative structures such as stems, roots, or leaves of the parent plant. This process allows plants to clone themselves, producing genetically identical offspring.

The process you are referring to is known as vegetative reproduction. It is a form of asexual reproduction in plants where a modified stem or root of a parent plant gives rise to new offspring that remain attached to the parent plant.

To get this answer, I analyzed the question and recognized the familiar terms "modified stem or root," "grows offspring," and "remain attached." By drawing on my knowledge of plant biology, specifically asexual reproduction, I was able to identify the correct term as vegetative reproduction.