Execute .NET Core 3.1 background worker tasks simultaneously
Async method out-of-the-box is using multiple threads for continuations. (Everything after await is continuation). Exception of this rule if existing SynchronizationContext. Don't Task.Run asynchronous I/O-bound operation, do it only for synchronous CPU-bound one. Asynchronous programming, SynchrinizationContext &amp; TaskScheduler.
Mixing .ContinueWith and await is not recommended as it doing the same but using different coding pattern. Mixing patterns with no reason makes the code more complicated.

If I were you, the code could look like this:
public class Worker : BackgroundService
{
    protected async override Task ExecuteAsync(CancellationToken stoppingToken)
    {
        // all methods will throw OperationCanceledException if cancellation was happened,
        // then checking if it was canceled is a kind of redundancy here.
        while (true) 
        {
            var task1 = MethodOneAsync(stoppingToken);
            var task2 = MethodTwoAsync(stoppingToken);
            await Task.WhenAll(task1, task2, Task.Delay(10000, stoppingToken)); // inlining Task.Delay here changes the logic: 10 seconds counted between starts of two iterations not between finish of previous and start of next.
        }
    }

    private async Task MethodOneAsync(CancellationToken stoppingToken)
    {
        Debug.WriteLine($&quot;MethodOneAsync started @ {DateTime.Now:dd.MM.yyyy HH:mm:ss.ffff}&quot;);
        await Task.Delay(1000, stoppingToken);
        Debug.WriteLine($&quot;MethodOneAsync finished @ {DateTime.Now:dd.MM.yyyy HH:mm:ss.ffff}&quot;);
    }

    private async Task MethodTwoAsync(CancellationToken stoppingToken)
    {
        Debug.WriteLine($&quot;MethodTwoAsync started @ {DateTime.Now:dd.MM.yyyy HH:mm:ss.ffff}&quot;);
        await Task.Delay(1000, stoppingToken);
        Debug.WriteLine($&quot;MethodTwoAsync finished @ {DateTime.Now:dd.MM.yyyy HH:mm:ss.ffff}&quot;);
    }
}

About Worker class purpose. It looks like Scheduler not Worker because it repeats some work continuously. Consider to change its name.