Hi Dhanashree,

Please find the code. I hope it will get you job done(Just Change the fields as per your need)
A trigger which will be fired whenever a contact is updated/Inserted and update the account and a batch you can either schedule or trigger on an ad-hoc request.

Note :- I have implemented custom logic where the batch will pick only those records which are updated after the last batch run date/Time (after the first time). This will ensure that redundant records are not processed.
For this logic to work you need to create a custom setting which will have a field to store the last run batch date/time.

Please mark this as best answer if it solves your issue.

Trigger :- updateCallOnAcctTrigger

trigger updateCallOnAcctTrigger on Contact (after insert,after update) {
    set<id> actId = new set<id>();
    
    for(Contact cnt: Trigger.new){
        if(Trigger.isInsert && Trigger.isAfter) 
        	actId.add(cnt.AccountId);
        if(Trigger.isUpdate && Trigger.isAfter && 
           (Trigger.NewMap.get(Cnt.Id).phone != Trigger.OldMap.get(Cnt.Id).phone)){
        	actId.add(cnt.AccountId);       
           }     
    }
    
    if(actId.size() > 0){
        updateAccountNoBatch uptAcct = new updateAccountNoBatch(actId);
        List<Account> updateActList = uptAcct.processAccount();
        if(updateActList.size() > 0){
            update updateActList;
        }
    }

}

// Initialize a class with modifier as Global. For a batch class the class needs to implement the interface Database.Batchable
// and Schedulable for the class to be scheduled.
Global class updateAccountNoBatch Implements Schedulable, Database.Batchable<sObject>{
    // Declare a Map for AccountId --> Contact Number
    Map<String,String> mapAccountsToProcess;
	
    // Field To store Last Batch Run
    Datetime lastRun = null;
    
    // Constructor called in batch Mode
    global updateAccountNoBatch(){
    	Customization_Settings__c cs = Customization_Settings__c.getOrgDefaults();
        if(cs.Batch_Last_Run_Time__c != null)
			lastRun = cs.Batch_Last_Run_Time__c;
    }
    
    // Constructor called from trigger
    global updateAccountNoBatch(Set<Id> AcctIds){
    	mapAccountsToProcess = new Map <String,String>();
        for(Contact ct: [Select Id, AccountId,Phone from Contact where AccountId In : AcctIds order by lastmodifiedDate Desc Limit 1]){
        	if(ct.Phone != null) 
            	mapAccountsToProcess.put(ct.AccountId, ct.Phone);   
        }
    }
    // The Database.batchable has three methods to be implemented. This will be same for all the batch classes.
    // Start Method Query the data
    global Database.QueryLocator start(Database.BatchableContext BC){
        String query = '';
        if(lastRun != null)
        	query = 'Select Id, AccountId,Phone'+
            		'from Contact where AccountId IN (Select Id,AccountNumber from Account)'+
            		'and LastModified >: lastRun order by lastModifieddate Desc Limit 1';
        else
            query = 'Select Id, AccountId,Phone from Contact where AccountId IN (Select Id,AccountNumber from Account) order by lastModifieddate Desc Limit 1';
        return Database.getQueryLocator(query);
    }
    
    // Main Logic executes in This method
    global void execute(Database.BatchableContext BC, List<Contact> scope){
        mapAccountsToProcess = new Map <String,String>();
        for(Contact ct : scope){
            if(ct.Phone != null)
            	mapAccountsToProcess.put(ct.AccountId, ct.Phone);   
        }
        
        if(mapAccountsToProcess.size() > 0 ){
            List <Account> listAccountsToUpdate	= processAccount();			
			if (listAccountsToUpdate.size() > 0)
                update listAccountsToUpdate;
        }
    }
    
    // When all the records are performed upon, this function is called.
    global void finish(Database.BatchableContext BC){
        
    }
    
    
    // This is called when a class is scheduled.
    global void execute(SchedulableContext ctx) {
		updateAccountNoBatch s = new updateAccountNoBatch();        
        database.executebatch(s,200);         	
	}	
    
    // Real Processing is taking place here.
    public List<account> processAccount(){
        List<account> returnAct = new List<Account>();
        
        for(String str : mapAccountsToProcess.keySet()){
            Account actNew = new Account(Id = str);
            actNew.Phone = mapAccountsToProcess.get(Str);
            returnAct.add(actNew);
        }
        
        if(returnAct.size() > 0){
            return returnAct;
        }
        
        return null;
    }
}