Contexts and parallelization

I am not 100% comfortable about this aspect of parallelization, so this is not going to be a full answer, but here are some hints:

In order for a piece of code to run in parallel, all definitions it uses must be copied to the subkernels. This is called "distributing" the definitions and can be done manually by DistributeDefinitions. It is also done automatically by most parallel functions (such as ParallelTable). A notable exception was ParallelEvaluate until version 10.4 .

Automatic distribution is done only for $Context. This is controlled by $DistributedContexts.

Why not distribute definitions for all contexts? Because for a package to work properly, it may not be sufficient to just copy all definitions from its context. The package may do things upon loading which are more complex than issuing definitions. E.g. it may load LibraryFunctions. Thus packages are meant to be loaded on parallel kernels using ParallelNeeds instead of just copying their definitions over from the main kernel.

What can you do then?

Either

DistributeDefinitions[const`precisionRate]

to manually distribute the definitions of this one symbol, or

$DistributedContexts := {$Context, "const`"}

to automatically distribute everything from the const` context. The second solution looks cleaner to me.

You need to SetSharedVariable on const`precisionRate. ParallelTable is not sharing this definition. Perhaps because it is not in the context of f.

Launch the kernels and then share.

LaunchKernels[]
SetSharedVariable[const`precisionRate]

Then ParallelTable works as expected.

ParallelTable[{y, f[y]}, {y, -1, 1, 0.1}]

Remember to UnsetShared.

UnsetShared[const`precisionRate]

Hope this helps.