This is just an initial thinking process related to your project:
6 HC-SR04 2 pins (trigger, echo) total 12 pins, 6 on will need to be support interrupt on change. But only RB4-RB7 support that (only 4), it is a little bit trickier in this way, because also you need to send the count to the LCD display (only for 6 spots)
There will be a way without using interrupts, but if you read the datasheet from HC-SR04 the min time is 150us and max time 25ms on the width of the echo signal (38ms without obstacles), then I think will be about 50ms per slot checking with a total of 300ms to check the 6 spots
My idea without using the interrupt on change behavior is using a timer interrupt, and the system will have three states for detecting: TRIGGER (10us), WAIT_ECHO(unknown time), ECHO_INIT, ECHO_END (up to 38ms)
Now suppose we feed the system with a 8MHz xtal, then will we have about 500ns or 0.5us per instruction (asm, not C), As you need to send the signal TRIGGER for 10us, the interrupt time need to be less than that number, I will suppose I want to have interrupts each 2.5us, then only 5 asm instructions per interrupts.
For a 20MHz xtal, then will we have about 200ns or 0.2us per instruction (asm, not C), this will give us about 12.5 instructions. We can increase from 2.5us to 5us to get 25 instructions executed between interrupts.
38ms/5us = 7600
Now add that also you need to write to the LCD the results, the LCD has their own times too.