Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$.  Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$.  Compute $x^2+y^2+z^2$.
1. We start by noting that the sum of the squares of the diagonals of a parallelepiped is equal to the sum of the squares of its side lengths. This can be derived using the Law of Cosines in three dimensions.

2. Let the parallelepiped be denoted as $ABCDEFGH$ with side lengths $AB = CD = FG = EH = x$, $AD = BC = EF = GH = y$, and $AF = BG = CH = DE = z$.

3. The space diagonals of the parallelepiped are $ACG$, $BDF$, $CEH$, and $DGB$. The lengths of these diagonals are given as $15$, $17$, $21$, and $23$.

4. The length of a space diagonal in a parallelepiped can be calculated using the formula:
   \[
   \sqrt{x^2 + y^2 + z^2}
   \]
   Therefore, we have the following equations:
   \[
   \sqrt{x^2 + y^2 + z^2} = 15
   \]
   \[
   \sqrt{x^2 + y^2 + z^2} = 17
   \]
   \[
   \sqrt{x^2 + y^2 + z^2} = 21
   \]
   \[
   \sqrt{x^2 + y^2 + z^2} = 23
   \]

5. Squaring each of these equations, we get:
   \[
   x^2 + y^2 + z^2 = 15^2 = 225
   \]
   \[
   x^2 + y^2 + z^2 = 17^2 = 289
   \]
   \[
   x^2 + y^2 + z^2 = 21^2 = 441
   \]
   \[
   x^2 + y^2 + z^2 = 23^2 = 529
   \]

6. Summing these equations, we get:
   \[
   4(x^2 + y^2 + z^2) = 225 + 289 + 441 + 529
   \]

7. Simplifying the right-hand side:
   \[
   225 + 289 + 441 + 529 = 1484
   \]

8. Therefore:
   \[
   4(x^2 + y^2 + z^2) = 1484
   \]

9. Dividing both sides by 4:
   \[
   x^2 + y^2 + z^2 = \frac{1484}{4} = 371
   \]

The final answer is $\boxed{371}$.