I would suggest using fluentvalidation, as it allows to separate and reuse validation rules.
In your case assuming the startdate is part of the CalendarAvailabilityRequest, you would add a validator for the request dto:
public class CalendarAvailabilityRequestValidator : 
AbstractValidator<CalendarAvailabilityRequest> 
{
  public CalendarAvailabilityRequestValidator() 
  {
    RuleFor(request => request.StartDate)
        .Must(BeAValidDateFormat).WithMessage("Date must follow the format: yyyy-mm-dd")
        .NotNull().WithMessage("A start date must be provided.");
  }
  // will only match yyyy-mm-dd
  private static bool BeAValidDateFormat(string date)
    => Regex.IsMatch(date, "2\d{3}\-(0[1-9]|1[012])\-(0[1-9]|[12][0-9]|3[01])$", RegexOptions.Compiled);
}
Within your controller you instanciate a validator and let it validate:
[Route("{providerName}/CalendarAvailability")]
[HttpGet]
public Task<IActionResult> GetCalendarAvailability(CalendarAvailabilityRequest request)
{
    var validationResult = new CalendarAvailabilityRequestValidator().Validate(request);
    if (!validationResult.IsValid)
    {
        Log.Warning(validationResult.Errors.ToString());
        return BadRequest(validationResult.Errors);
    }
    var statDate = DateTime.ParseExact(request.StartDate, "yyyy-mm-dd", CultureInfo.InvariantCulture);
    //TODO: calendar availability logic
    return OK(); 
}
Of course you can just as well use the regex from above and validate the request wihin your controller.
Another option is to try catch using DateTime.ParseExact something like this:
try
{
    var statDate = DateTime.ParseExact(request.StartDate, "yyyy-mm-dd", CultureInfo.InvariantCulture);
}
catch(exception ex)
{
  Log.Warning("Request for {startdate} was invalid: {message}", request.StartDate, ex.Message);
  return BadRequest(ex.message);
}
But I would recomment to avoid try catch when you can validate the input, unless you really need to.