PowerShell - How to make defined constants and functions accessible in ForEach-Object -Parallel

I'm working on a PowerShell script that needs to use ForEach-Object -Parallel to run several iterations of a process simultaneously. Here's my code (with names changed to protect the innocent). Right now, I have essentially a copy of the function in the loop by converting it to a string (I found that in another answer here on SE).

If I try to use a constant in one of these functions, it comes up empty, even though every constant I have is declared in the highest scope. Also, my "ProcessItem" function doesn't recognize any other declared functions and errors out if I try to call one of them.

I'm using Powershell 7.5, and I don't have any errors in the constants or functions themselves (which I have verified by running each one in its own environment). If I comment out the calls to other functions and replace the constants with hard-coded values, the program runs all the way through without dying.

# Number of seconds between the process
Set-Variable -Name WAIT_TIME -Option Constant -Value 5

$ProcessItemString = ${function:ProcessItem}.ToString()

0..($allItems.Count - 1) | ForEach-Object -Parallel {
    ${function:LocalProcessItem} = $using:ProcessItemString

    $a, $b = LocalProcessItem -itemID ($using:allItems)[$_].item_id -category ($using:allItems)[$_].category -requests $using:requests

    echo $a
    echo $b
} -Throttle 20

So here's my question: Is there a way to make those constants and functions accessible inside of these other threads?

I'm more concerned about the function calls than the constants, but if there's a way to make them invokable without just making them a subfunction, that would be preferrable.

EDIT: I don't know if this affects the answer at all, but this process will be going through 2000 items, and I will be setting the throttle value higher before actually running the whole thing.
I've figured out the constants, but not the functions yet.