How to inherit base class dependencies without redefining theme in child class service arguments?
The question asks:

Is there any way to inject the Base Class service , when defining a
child class service (NodeHandler , ParagraphHandler) without using the
arguments: part in the service definition

Yes. For example, look in core.services.yml to see how this is done in core Drupal. Most core Drupal plugin managers have a parent service named default_plugin_manager. The default_plugin_manager service is defined like this:
  default_plugin_manager:
    abstract: true
    arguments: ['@container.namespaces', '@cache.discovery', '@module_handler']

As you can see, there are three services injected into the default_plugin_manager service. Because it is declared as &quot;abstract&quot;, this service definition is more like an alias or a placeholder - it's simply there to define a set of arguments. Note there's no class associated with this service.
A concrete plugin manager, for example the plugin.manager.block service, is then defined like this:
  plugin.manager.block:
    class: Drupal\Core\Block\BlockManager
    parent: default_plugin_manager
    arguments: ['@logger.channel.default']

This service has FOUR arguments injected; the three defined by the parent default_plugin_manager plus the addition argument defined here.
When you look at the concrete class BlockManager that implements this service, you see that BlockManager has a constructor that looks like this:
public function __construct(\Traversable $namespaces, CacheBackendInterface $cache_backend, ModuleHandlerInterface $module_handler, LoggerInterface $logger) {
...
}

As you can see, this constructor takes four arguments. The first three defined by the default_plugin_manager service and the last one defined by the plugin.manager.block service.
Core Drupal defines a DefaultPluginManager base class which is extended by BlockManager and other plugins. BlockManager accepts all four arguments in its constructor. Note that if the parent (DefaultPluginManager in this case) has a constructor then you need to call that from the subclass constructor. This way many different plugin managers can extend DefaultPluginManager and inherit those three injected services.
In your specific case, your services.yml file might look like this:
  my_parent_service:
    abstract: true
    arguments: ['@entity_type.manager', '@request_stack', '@sacd_user.permissions', '@logger.factory']

  sacd_rest_api.node_handler:
    class: Drupal\sacd_rest_api\service\NodeHandler
    parent: my_parent_service
  sacd_rest_api.parapgraph_handler:
    class: Drupal\sacd_rest_api\service\ParagraphHandler
    parent: my_parent_service

You do not need a create() method anywhere, and you do not need to use ControllerBase at all.