Anyway, if the system (server/network/storage) can handle the load, it would be worth doing faster backups.
That is exactly my point: the backup process will always saturate on those three things you and I mentioned and therefore running in parallel will not be faster. If you have a "simple" system with one pool of local storage (all VMs on the same physical disks) and one network share to copy to, parallelism will not help (multiple local pools, multiple NICs, multiple LACP ports with multiple network storages on different servers - I'm with you!):
If you cannot read from your local disk fast enough, doing it in parallel will not be faster either. In case of hard disk, it'll slow down significantly
If you saturate your network bandwidth, you will not be faster either.
If your CPU is not able to compress fast enough, use a less demanding compression scheme or no compression at all. In the later case, you're never CPU bound, always disk and network. Normally LZOP compression is transparent in modern CPUs.
The CPU part is the only point I can see where parallelism could be an advantage if you choose to use a demanding compression scheme. It is debatable if you could transfer your VMs faster with higher compression rate in parallel than with a streamlined compression. This reminds of questions like having a trained dog deliver harddisks or upload them via network to your destination :-D
And yes, if you have your big system with 24x NVME and dual or quadruple 100 GBE, your CPU is most certainly your bottleneck with compression, but seriously, with that kind of money, you'd go straight for a real shared storage cluster solution.
The workaround for this is typically to make the new server join the older server in a cluster and migrate one machine at a time.(which is way slower than the all-in-one approach) and then delete/detach the older server.
Yes, slower, but less application downtime - you can even do it live in normal office hours. I never met a customer who would not prefer one-by-one migration during office hours.
(If you have a single server and the potential of migrating it to another system, just go with ZFS ... you have incremental replication and you're really fast migrating VMs - no need for backup&restore on migration with ZFS.)