The difference between a $P_1$ partition and a $P_2$ partition is that a $P_1$ partition cannot contain even numbers congruent to $-2$ (mod $4r$), while a $P_2$ partition cannot repeat odd numbers. So to find a bijection, our first hope would be that these "excluded partitions" somehow map onto each other.
The additional fact we have about both $P_1$ and $P_2$ partitions is that odd parts are always congruent to -1 (mod $2r$). $~$ Interestingly, these values are exactly 1/2 of the values which are excluded from $P_1$ partitions.
So, how can we convert between repeated odd numbers and even numbers twice their size? Well, we can think of the even numbers as pairs of odd numbers.
This bijection in fact maps $P_1$ and $P_2$ onto each other quite cleanly.
For $r=25$, we have that odd parts (in both $P_1$ and $P_2$) must always end with "49" or "99", and in $P_2$ the odd parts cannot be repeated, while in $P_1$ there are no parts ending in "98". So, where $P_1$ has repetitions of odd numbers, we pair them up (leaving at most one unpaired), yielding numbers ending in "98" for the $P_2$ partition. In the other direction, we split every $P_2$ number ending in "98" into two parts of half the size to get a $P_1$ partition.