Concurrency is a property of algorithm. Parallelism is a property of the running environment. The hope is given that you have large amount of concurrency, that concurrency would be reasonably and easily distributed over parallel execution units (CPU, sockets+select loops, goroutines, tasks, processes, separate machines, VMs etc...).
So if you have concurrency in your problem domain/algorithm ( say web page requests for example ), and, your language has reasonable ways to handle it. Abstractions like processes, tasks, etc, you can make each request spawn an isolated, lightweight Erlang process (it takes just microseconds and only a few Ks of memory). Then finally at runtime, if you have a multi-core machine or started your Erlang VM with a good number of async IO threads, there is a good chance that you'll get good parallelism too! But if you only run that bytecode on a single core machine, you might not get parallelism but concurrency will still be there.
> Some resource has to be under contention,
Why? That is just resource contention. You don't want resource contention if you can avoid it. Think of concurrent as independent units of executions.
> In Erlang, you have to ship everything to everyone, but the resources are still technically shared...just very inefficiently.
Because it keeps things (including faults, but also logical decomposition) simple. Also it maps well to real life. You have to send messages to other people, emails, phone, image. You don't directly plug a shared network back-plane into their brain. So when that crashes the whole group dies. You want to send a message to them and then continue minding your own business.
