To solve the given problem, we need to find all prime numbers $p, q, r, k$ such that the equation

$$pq + qr + rp = 12k + 1$$

is satisfied.

### Step-by-Step Solution

1. Analyzing the Equation:

The equation can be written as:

$$pq + qr + rp = 12k + 1$$

Our goal is to find integer values (specifically prime numbers) for $p, q, r,$ and $k$ that satisfy the equation.

2. Evaluating Possibilities:

Since $p, q, r$ are prime numbers, let's start with small prime numbers and test them.

3. **Setting $p = 2$**:

- Substituting $p = 2$, we get:

$$2q + qr + 2r = 12k + 1$$

- Rearranging, it becomes:

$$qr + 2(q + r) = 12k + 1$$

Let's test with $q = 3$ (the next smallest prime number):

4. **Setting $q = 3$**:

- Substituting $q = 3$ into the equation, we have:

$$3r + 2(3 + r) = 12k + 1$$

- Simplifying, we find:

$$3r + 6 + 2r = 12k + 1 \implies 5r + 6 = 12k + 1 \implies 5r = 12k - 5$$

- Therefore, $5r \equiv 7 \pmod{12}$. Possible values of $r \equiv 7 \pmod{12}$ need to be prime.

5. **Trying Possible Values for $r$**:

- Let’s check while satisfying $5r \equiv 7 \pmod{12}$:

- $r = 11$: when $r = 11$, check:

$$5(11) + 6 = 55 + 6 = 61$$

Check if $61 = 12k + 1$:

$$61 - 1 = 60, \, \text{then } k = \frac{60}{12} = 5$$

- Therefore, $(p, q, r, k) = (2, 3, 11, 5)$.

6. Considering Permutations:

Due to symmetry in the equation $pq + qr + rp$, permuting $(p, q, r)$ gives valid solutions without changing the form of the equation:

- Therefore, the valid solutions are:

$$(p, q, r, k) = (2, 3, 11, 5); (2, 11, 3, 5); (3, 11, 2, 5); (3, 2, 11, 5); (11, 2, 3, 5); (11, 3, 2, 5)$$

7. Conclusion:

Thus, all prime number solutions $(p, q, r, k)$ such that the given equation holds are:

$$\boxed{(2, 3, 11, 5); (2, 11, 3, 5); (3, 11, 2, 5); (3, 2, 11, 5); (11, 2, 3, 5); (11, 3, 2, 5)}$$