We are tasked with finding all triples of primes $(p, q, r)$ that satisfy the equation:

$$3p^4 - 5q^4 - 4r^2 = 26.$$

We begin by analyzing the equation with respect to the properties of prime numbers.

1. Testing Small Primes:
- Since $p^4$, $q^4$, and $r^2$ grow rapidly for primes larger than a few small values, it's practical to test small primes for possible values of $p$, $q$, and $r$.

2. **Case $p = 5$**:
- Assume $p = 5$, substitute into the equation:
$$3(5)^4 - 5q^4 - 4r^2 = 26.$$
$$3 \times 625 - 5q^4 - 4r^2 = 26.$$
$$1875 - 5q^4 - 4r^2 = 26.$$
$$5q^4 + 4r^2 = 1849.$$

3. **Testing $q = 3$**:
- Assume $q = 3$, substitute into the equation:
$$5(3)^4 + 4r^2 = 1849.$$
$$5 \times 81 + 4r^2 = 1849.$$
$$405 + 4r^2 = 1849.$$
$$4r^2 = 1444.$$
$$r^2 = 361.$$
$$r = 19.$$

4. Verification:
- Verify the initially assumed and calculated primes $(p, q, r) = (5, 3, 19)$:
$$3 \times 5^4 - 5 \times 3^4 - 4 \times 19^2 = 26.$$
Calculate each term:
$$3 \times 625 = 1875, \quad 5 \times 81 = 405, \quad 4 \times 361 = 1444.$$
$$1875 - 405 - 1444 = 26.$$

Thus, the only triple of primes $(p, q, r)$ that satisfies the equation is:

$$\boxed{(5, 3, 19)}.$$