a form of asexual reproduction in plants in which a modified stem or root of the parent organism grows offspring that stay attached

The form of asexual reproduction you are describing is called vegetative propagation. It involves the growth of new individuals from specialized plant structures, such as stems or roots, of the parent plant. These specialized structures are known as vegetative propagules. They can develop adventitious roots and shoots, ultimately forming new plants while remaining attached to the parent organism. Some examples of vegetative propagation include runners in strawberry plants, tubers in potatoes, and rhizomes in ginger.