I managed to format the code.

It serves as a good example of something that should be in an attached notebook, so people looking for replies or making replies will not have to scroll and scroll. (I hope that did not sound harsh. Just a suggestion for next time.)

I don't believe your code is doing what you intended.

There is no definition of T0, so I made up one.

T0 = Table[x^int, {int, 1, 3^4}]
pret00 = Table[T0[[int]]*LegendreP[0, Cos[x]]*Sin[x], {int, 1, 3^4}]

Now to check and see if Integrate is doing something sensible:

Integrate[pret00[[1]], {x, 0, Pi}]

Yes, it is. So, something like this gives sensible results

ParallelTable[Integrate[pret00[[int]], {x, 0, Pi}], {int, 1, 12}]

or

ParallelTable[Integrate[pret00[[int]], {x, 0, Pi}], {int, 1, 3^4}]

which will produce fairly long output, but you can view it all if you wish.

However, I believe the Table that you have inside ParallelSubmit isn't doing what you expected because you are looking for the 0 index of a list, which will be "List".

ParallelTable[Integrate[pret00[[int]], {x, 0, Pi}], {int, 0, 1}]

A quick look at the ParallelSubmit documentation shows that the result of ParallelSubmit is a List of objects which are ready for computation on the kernels, but computation doesn't start until WaitAll or WaitNext is executed