Okay,

So a few things I want to point out..

The trigger will have to be on the child object Resume_Subscriptions__c as that is where insert and update will happen

The best way that I can think of is to use aggreate queries, as the trigger can be fired for multiple records with different contacts.

The trigger will also have to fire on delete of a child record.

Since it is a lookup field, you will also need to take care of situation where the contact alignment changes for a Resume_Subscriptions__c

Basically the idea is to

1. Iterate through trigger.new and collect all the Contact IDs in a set 'ContSet'

2. Issue an aggregate query to calculate the count of child records which are to the contact avaialble in ContSet.

3. Work with the query results to store the contents in a map of contact id, count..

4. Use this map to update the field value of contact.

Please use the below code. You might have to tweak it to work for you.

trigger RollUp on Resume_Subscriptions__c (after insert, after update, before delete) {

    // Set to hold the IDs of the Contacts of the Subscriptions
    Set<ID> ContactIDs= new Set<ID>();
    // Set to hold the IDs of the Subscriptions
    Set<ID> SubIDs = new Set<ID>();
    // Set to hold the IDs of the Contacts where the values need to be reset
    Set<ID> ContactIDsForReset = new Set<ID>();

    if(trigger.isupdate || trigger.isInsert)
    {
        for(Resume_Subscriptions__c rsSub : trigger.new )
        {
            // If the contact of a subscription has changed, then the old contact also needs to be updated 
            if( rsSub.Contact__c != trigger.oldmap.get(rsSub.ID).Contact__c)
                if( trigger.oldmap.get(rsSub.ID).Contact__c != null )
                    ContactIDs.add(trigger.oldmap.get(rsSub.ID).Contact__c);
            if(rsSub.Contact__c!=null)
            {
                ContactIDs.add(rsSub.Contact__c);
            }
        }
        ContactIDsForReset.addAll(ContactIDs);

        // The list of Contacts where the update has to take place
        List<contact> ContactsToUpdate = new List<contact>();    

        // Querying the database to get the Number of Subscriptions under an contact and the sum of the deal values of the Subscriptions under an contact
        AggregateResult[] Subscriptionsum = [select count(ID) NoOfSubscriptions, Contact__c ContactID from Resume_Subscriptions__c where Contact__c in: ContactIDs group by Contact__c ];
        
        for( AggregateResult ContactDetail : Subscriptionsum )
        {
            contact ContactToUpdate = new contact(ID = (ID)(ContactDetail.get('ContactID')));
            ContactToUpdate.No_of_Subscriptions__c = integer.valueOf(ContactDetail.get('NoOfSubscriptions'));
            ContactsToUpdate.add(ContactToUpdate);
            // The contact being handled should not be reset. Hence removing it from the ResetSet
            ContactIDsForReset.remove((ID)(ContactDetail.get('ContactID')));
        }
        
        for( ID i : ContactIDsForReset )
        {
            // Resetting the summed up values to 0 if the contact no longer has a subscription.
            contact ContactToUpdate = new contact( ID = i );
            ContactToUpdate.No_of_Subscriptions__c = 0;
            ContactsToUpdate.add(ContactToUpdate);
        }
        
        if(ContactsToUpdate.size() > 0)
            update ContactsToUpdate;
            
        ContactsToUpdate.clear();
	}

    if(trigger.isdelete)
    {
        for(Resume_Subscriptions__c rsSub : trigger.old )
        {
            if( rsSub.Contact__c != null)
                ContactIDs.add(rsSub.Contact__c);
            SubIDs.add(rsSub.ID);
        }
        ContactIDsForReset.addAll(ContactIDs);

        // The list of Contacts where the update has to take place
        List<contact> ContactsToUpdate = new List<contact>();    
        
        // Querying the database to get the Number of Subscriptions under an contact excluding the Subscriptions being deleted
        AggregateResult[] Subscriptionsum = null;
        if(ContactIDs.size() > 0)
        {
            Subscriptionsum = [select count(ID) NoOfSubscriptions, Contact__c ContactID from Resume_Subscriptions__c where Contact__c in: ContactIDs and ID not in: SubIDs group by Contact__c ];

            for( AggregateResult ContactDetail : Subscriptionsum )
            {
                contact ContactToUpdate = new contact(ID = (ID)(ContactDetail.get('ContactID')));
                ContactToUpdate.No_of_Subscriptions__c = integer.valueOf(ContactDetail.get('NoOfSubscriptions'));
                ContactsToUpdate.add(ContactToUpdate);
				// The contact being handled should not be reset. Hence removing it from the ResetSet
                ContactIDsForReset.remove((ID)(ContactDetail.get('ContactID')));
            }

            for( ID i : ContactIDsForReset )
            {
                // Resetting the summed up values to 0 if the contact no longer has a subscription.
                contact ContactToUpdate = new contact( ID = i );
                ContactToUpdate.No_of_Subscriptions__c = 0;
                ContactToUpdate.Total_Sum_of_Contract_Value__c = 0;
                ContactToUpdate.Contract_Billing_Retention_Status__c = 0;
                ContactsToUpdate.add(ContactToUpdate);
            }
        }
        
        if(ContactsToUpdate.size() > 0)
            update ContactsToUpdate;

        ContactsToUpdate.clear();
    }
}

Resume_Subscriptions__c