The first thing I think is that $p_1p_2p_3p_4$, which is a multiple of many $p_1$s,$p_2$s, $p_3$s, and $p_4$s, is certainly going to be a lot bigger than $p_1+p_2+p_3+p_4$.
Indeed. $p_1p_2p_3p_4 = (p_1p_2p_3 -1)p_4 + p_4=$
$(p_1p_2p_3-1)(p_4-1) + p_1p_2p_3 -1 + p+4 =$
$(p_1p_2p_3-1)(p_4-1) + (p_1p_2 -1)p_3 -1 + p_3 + p_4=$
$(p_1p_2p_3-1)(p_4-1)+(p_1p_2-1)(p_3-1) +p_1p_2 -2 +p_3+p_4=$
$(p_1p_2p_3-1)(p_4-1)+(p_1p_2-1)(p_3-1) + (p_1-1)p_2 -2 + p_2+p_3+p_4=$
$(p_1p_2p_3-1)(p_4-1)+(p_1p_2-1)(p_3-1) + (p_1-1)(p_2-1) + p_1 -3 p_2+p_3+p_4=$
$(p_1p_2p_3-1)(p_4-1)+(p_1p_2-1)(p_3-1) + (p_1-1)(p_2-1) -3+ (p_1 + p_2+p_3+p_4)\ge $
$(8-1)(2-1) + (4-1)(2-1)+(2-1)(2-1)-3+ (p_1 + p_2+p_3+p_4)=$
$11+ (p_1 + p_2+p_3+p_4)$.
Now we want $p_1+p_2+p_3+p_4 = p_1p_2p_3p_4 -15 =(p_1p_2p_3-1)(p_4-1)+(p_1p_2-1)(p_3-1) + (p_1-1)(p_2-1) -3+ (p_1 + p_2+p_3+p_4)-15$
or in other words
$(p_1p_2p_3-1)(p_4-1)+(p_1p_2-1)(p_3-1) + (p_1-1)(p_2-1)=18$.
Note: $p_1p_2p_3-1 \ge 7$ so $p_4-1 \le 2$. And $p_4-1 \ge 1$ so $p_1p_2p_3-1< 17$.
The tells us that $p_4 = 2,3$ and at least two of $p_1,p_2,p_3=2$.
If $p_4 = 3$ the only option is $p_1=p_2=p_3=2$.
If $p_4 =2$ we can have $p_1p_2p_3-1=7, 11$.
We can brute force this out, but as it was arbitrary which terms we factored out first. We can assume without loss of generality that $p_4\ge p_3,p_2,p_1$. So we have either $p_3=3$ and $p_1=p_2=p_3=2$.
Or that $p_3=2$ and all the primes are $2$. Butbut... $2+2+2+2\ne 2*2*2*2-15$ so $(2,2,2,3)$ is the only solution.
==== old answer below ====
If we assume $p_1 \le p_2 \le p_3\le p_4$ then
$p_1+p_2+ p_3 + p_4 = p_1p_2p_3p_4-15$
$15 = p_1p_2p_3p_4 - p_1-p_2-p_3-p_4=$
$p_1(p_2p_3p_4 -1) - p_2 - p_3-p_4=$
$(p_1-1)(p_2p_3p_4-1) + (p_2p_3p_4 -1)- p_2 - p_3-p_4=$
$(p_1-1)(p_2p_3p_4-1)+ (p_3p_4-1)p_2 -1-p_3-p_4=$
$(p_1-1)(p_2p_3p_4-1)+ (p_2-1)(p_3p_4-1) + p_3p_4 - 2 - p_3 - p_4=$
$(p_1-1)(p_2p_3p_4-1)+ (p_2-1)(p_3p_4-1) + (p_4-1)p_3 -p_4 -2=$
$(p_1-1)(p_2p_3p_4-1)+ (p_2-1)(p_3p_4-1) + (p_3-1)(p_4-1) -3$
So
$12 = (p_1-1)(p_2p_3p_4-1)+ (p_2-1)(p_3p_4-1) + (p_3-1)(p_4-1)$
Now $p_2p_3p_4 - 1\ge 7$ so $p_1-1 < 2$ so $p_1-1 =1$ and $p_1 = 2$.
And $p_2p_3p_4-1 \ge 7$ so
$5 \ge (p_2-1)(p_3p_4-1) + (p_3-1)(p_4-1)$ and $p_3p_4-1 \ge 3$ so $p_2-1 < 2$ and $p_2 = 2$.
And as $p_3p_4-1 \ge 3$ we have
$2\ge (p_3-1)(p_4-1)$.
If $p_3-1\ge 2$ then $p_4-1 \ge 2$ and that'd imply $2\ge 4$ which is a contradiction, so $p_3-1=1$ and $p_3 = 2$
And with that we have
$2+2+2+p_4 = 8p_4 -15$
$21 = 7p_4$
$p_4 =3$.
So that's the only (up to order) solution. $p_1,p_2,p_3,p_4 = 2,2,2,3$