Can I access parent Account info from a Case before Insert trigger?

David - Try the below code to get the lookup field values in the before insert operation

trigger MyTrigger on Case(before insert) {
    Set<Id> AccountIds = new Set<Id>();    
    for (Case cs: Trigger.new) {
        if (cs.AccountId != null) {
            AccountIds.add(cs.AccountId);
        }
    }
    Map<Id, AccountId> accountEntries = new Map<Id, Account>(
        [select Id,Name from Account where id in :AccountIds]
    );
for (Case caseRec : Trigger.new) {
  String accountName = accountEntries.get(caseRec .AccountId).Name; //Getting Account name, similarly you can get other field value as well 
        //your logic
    }
}