Let's start with what's known. If the P MOSFETs (Q5 and Q6) are repeatedly failing, it's most likely the circuit is exceeding the V-gs (max) rating for the parts. For these MOSFETs, V-gs (max) is +/-16V. For your circuit (without R9, R10), that means the gate is seeing over 40V (not likely) or less than 8V. Notice that Q9, Q10 are configured as switches (good), whereas Q1, Q2 are configured as emitter followers (bad!). When Q3, Q4 are OFF, there is no path for base current for Q1, Q2 so they're OFF and the gates of Q5, Q6 are pulled to +24V through R7, R8, V-gs = 0, and Q5, Q6 are OFF. So far so good. However, when Q3, Q4 turn ON. This pulls the base of Q1, Q2 toward ground, forward biasing the collector-base junction. I'm not certain, but I suspect this zaps Q1, Q2, leaving only the emitter-base junction intact, looking like a diode between R1-R7, R2-R8. So you now have a resistor divider between +24 and ground consisting of R7, R1, the Q2 forward biased base-emitter junction, and the Vce-sat of Q4. If you do the arithmetic, this pulls the gate of Q5 down to almost 8V, which from the MOSFET spec is VERY BAD.
If the above is correct, the reason the circuit works (even with Q1, Q2 zapped) with the addition of R9, R10 is that they create another divider with R1, R2 that limits the minimum gate voltage on Q5, Q6 to something around 20V rather than 8V, well within spec.
One solution would be to reconfigure Q1, Q2 to be switches like Q9, Q10, with the emitters tied directly to +24V; R7, R8 connected between the collectors and +12V; and R9, R10 left in the circuit like R14. With this configuration, the gates of Q5, Q6 would switch between +24V and +12V, a 12 volt swing, well within the V-gs (max) spec.
One other point to keep in mind is the sequence you use with your PIC microcontroller to change motor direction. Be careful not to allow both MOSFETs on one side of the H-bridge to be on simultaneously, even for a short time, since this creates a short between +24V and ground. The sequence of instructions should be: Q5,Q8 ON, Q6,Q7 OFF -> Q5 OFF -> Q7 ON -> insert braking delay -> Q8 OFF -> Q6 ON.
Hope this helps.