Let $T$ be a right triangle with sides having lengths $3$, $4$, and $5$. A point $P$ is called [i]awesome[/i] if P is the center of a parallelogram whose vertices all lie on the boundary of $T$. What is the area of the set of awesome points?
1. **Identify the vertices of the right triangle \( T \):**
   The right triangle \( T \) has vertices at \( (0,0) \), \( (3,0) \), and \( (0,4) \).

2. **Understand the condition for a point \( P \) to be awesome:**
   A point \( P \) is awesome if it is the center of a parallelogram whose vertices lie on the boundary of \( T \). 

3. **Determine the possible locations of \( P \):**
   Since \( P \) is the center of a parallelogram, it must be the midpoint of the diagonals of the parallelogram. The vertices of the parallelogram must lie on the boundary of the triangle \( T \).

4. **Analyze the constraints on the parallelogram:**
   The vertices of the parallelogram must lie on the segments \( (0,0) \) to \( (3,0) \), \( (0,0) \) to \( (0,4) \), and \( (3,0) \) to \( (0,4) \). 

5. **Consider the possible shapes of the parallelogram:**
   The parallelogram can be formed by choosing two points on one leg of the triangle and two points on another leg. For example, if we choose two points on the segment \( (0,0) \) to \( (3,0) \) and two points on the segment \( (0,0) \) to \( (0,4) \), the center \( P \) will lie within a certain region.

6. **Calculate the area of the set of awesome points:**
   The set of awesome points forms a region within the triangle. By symmetry and geometric properties, this region is a smaller triangle within the original triangle \( T \).

   - The vertices of this smaller triangle are the midpoints of the legs of the original triangle.
   - The midpoint of \( (0,0) \) to \( (3,0) \) is \( (1.5, 0) \).
   - The midpoint of \( (0,0) \) to \( (0,4) \) is \( (0, 2) \).
   - The midpoint of \( (3,0) \) to \( (0,4) \) is \( (1.5, 2) \).

7. **Determine the area of the smaller triangle:**
   The smaller triangle has vertices at \( (1.5, 0) \), \( (0, 2) \), and \( (1.5, 2) \).

   - The base of the smaller triangle is the distance between \( (1.5, 0) \) and \( (1.5, 2) \), which is 2 units.
   - The height of the smaller triangle is the distance between \( (0, 2) \) and the line \( x = 1.5 \), which is 1.5 units.

   The area of the smaller triangle is given by:
   \[
   \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 1.5 = 1.5
   \]

The final answer is \(\boxed{\frac{3}{2}}\).