Hi Tanvi,
I already typed my solution half-way through when I realised you cannot do this directly. You will most definitely need a lookup relationship bewteen Parent__c and Teacher__c. The reason for this is that you have to build maps with a key (that should at best be unique) so you can use the map to get parent records to update when updating Teacher__c records e.g. 

Map<String, Parent__c> parentData = new Map<String, Parent__c>();

So now what will you use as the key to the Map above? You can't use the Teacher Name because it may change when the Teacher record is updated. So you need a lookup on the Teacher record which points to the parent. Or you need to fake this if for some reason you do not or cannot add a lookup on the Teacher object to point to the parent.

So how do you fake it?

You could create a field on the parent object and put the Teacher record Id in this field. What this means is that, i am assuming whenever you create a teacher record, you create a parent and copy the name, expertise and the Id into the relevant fields.

This said you could then do the following;

for(Parent__c p : [SELECT Teacher_Name__c, Teacher_Expertise__c, TeacherId FROM Parent__c]){
     parentData.put(TeacherId, p);
}

2.) Go over the list of Teacher records you are updating and update the parent records, preferable in an after insert/update trigger on the Teacher__c object. Use a list to collect all parent records you will like to update. 
List<Parent__c> parentRecordsToUpdate = new List<Parent__c>();
for(Teacher__c t : teacherRecords){
    if(parentData.get(t.TeacherId) != null){
       parentData.get(t.TeacherId).Teacher_Name__c = t.Teacher_Name__c;
       parentData.get(t.TeacherId).Teacher_Expertise__c = t.Teacher_Expertise__c;
       parentRecordsToUpdate.add(parentData.get(t.TeacherId));
    }
}

try{
    update parentRecordsToUpdate;
} catch(Exception e){
    System.debug('An error occured while updating the parents');
}

Hope this comes close to what you want. If not I hope you get some ideas on how to proceed.

Cheers