Final answer:
Gap junctions in animal cells and plasmodesmata in plant cells provide direct intercellular communication by allowing small molecules to pass between neighboring cells.
Step-by-step explanation:
The part of the cell that directly permits intercellular communication with adjacent cells is the gap junction in animal cells and the plasmodesmata in plant cells. These structures form connections between the plasma membranes of neighboring cells, allowing for the passage of small signaling molecules like calcium ions (Ca2+). The presence of these channels enables cells to quickly transmit signals to one another and coordinate their responses, maintaining the individuality of each cell while supporting their ability to communicate as part of a larger network.