As stated in DRF documentation http://www.django-rest-framework.org/api-guide/parsers/#multipartparser, in order to parse multipart/form-data, the MultiPart and form parser must be used. I have a supiscion this is a problem in the Django Rest Framework because I saw a solution on their github issue saying it work using APIView.
from django.contrib.auth.models import User
from rest_framework import viewsets
from api.serializers import UserSerializer,
from rest_framework.parsers import MultiPartParser, FormParser
from rest_framework.response import Response
from rest_framework import status
class UserViewSet(viewsets.ModelViewSet):
"""
API endpoint that allows users to be viewed or edited.
"""
queryset = User.objects.all()
parser_classes = (MultiPartParser, FormParser)
serializer_class = UserSerializer
Picture of me sending request to Postman with result:

Edit: Adding UserSerializer class
class UserSerializer(serializers.HyperlinkedModelSerializer):
snapcapsules = SnapCapsuleSerializer(
many=True,
read_only=True,
allow_null=True,
)
class Meta:
model = User
fields = ('snapcapsules', 'url', 'username', 'email', 'password', )
write_only_fields = ('password',)
def create(self, validated_data):
user = User.objects.create(
username=validated_data['username'],
email=validated_data['email'],
)
user.set_password(validated_data['password'])
user.save()
return user
def update(self, instance, validated_data):
capsules_data = validated_data.pop('snapcapsules')
for capsule in capsules_data:
SnapCapsule.objects.create(user=instance, **capsule)
return instance



UserSerializerplease.