How to merge .fastq.qz files into a single .fastq.gz with their same id without losing any content in parallel
All you need is cat. You won't find any better tool for a simple job like this. Just run:

cat NA24694_GCCAAT_L001_R1*fastq.gz &gt; EA00694_GCCAAT_L001_R1.fastq.gz
cat NA24694_GCCAAT_L001_R2*fastq.gz &gt; EA00694_GCCAAT_L001_R2.fastq.gz
cat NA24694_GCCAAT_L001_R3*fastq.gz &gt; EA00694_GCCAAT_L002_R1.fastq.gz
cat NA24694_GCCAAT_L001_R4*fastq.gz &gt; EA00694_GCCAAT_L002_R2.fastq.gz


That should work just fine. You say you tried cat and it "didn't work" but since you don't tell us how it failed, I can't really help. The only issues I can think of is that either there are too many files (as in several hundred thousand, whatever the value returned by getconf ARG_MAX on your system which is 2097152 on mine) or, more likely, you are running out of disk space. 

If it's a disk space issue, you might be able to get around it by adding each file and then deleting it:

for i in 1 2 3 4; do 
    for file in NA24694_GCCAAT_L001_R${i}_*fastq.gz; do
        cat "$file" &gt;&gt; EA00694_GCCAAT_L001_R${i}.fastq.gz &amp;&amp; rm "$file"
    done
done


IMPORTANT: the command above is destructive. It will delete each file after it has been added to the new one. If everything works fine that's not a problem, but if I've made a mistake or your file names are slightly different you might lose your data. So I strongly urge you to simply run the cat commands in the first section on a machine with enough disk space to deal with them.