Example: Major version upgrade from 1.1.1.0 to 1.2.0.2 with a mix of default and custom parameter groups

Find the DBCluster that you want to upgrade, and the template you used to create it. For example:

Description: Base Template to create Neptune Stack with Engine Version 1.1.1.0 using custom Parameter Groups 
Parameters:
  DbInstanceType:
    Description: Neptune DB instance type
    Type: String
    Default: db.r5.large
Resources:
  NeptuneDBClusterParameterGroup:
    Type: 'AWS::Neptune::DBClusterParameterGroup'
    Properties:
      Family: neptune1
      Description: 'NeptuneDBClusterParameterGroup with family neptune1'
      Parameters:
        neptune_enable_audit_log: 0
  NeptuneDBParameterGroup:
    Type: 'AWS::Neptune::DBParameterGroup'
    Properties:
      Family: neptune1
      Description: 'NeptuneDBParameterGroup with family neptune1'
      Parameters:
        neptune_query_timeout: 20000
  NeptuneDBCluster:
    Type: 'AWS::Neptune::DBCluster'
    Properties:
      EngineVersion: 1.1.1.0
      DBClusterParameterGroupName:
        Ref: NeptuneDBClusterParameterGroup
    DependsOn:
      - NeptuneDBClusterParameterGroup
  CustomNeptuneDBInstance:
    Type: 'AWS::Neptune::DBInstance'
    Properties:
      DBClusterIdentifier:
        Ref: NeptuneDBCluster
      DBInstanceClass:
        Ref: DbInstanceType
      DBParameterGroupName:
        Ref: NeptuneDBParameterGroup
    DependsOn:
      - NeptuneDBCluster
      - NeptuneDBParameterGroup
  DefaultNeptuneDBInstance:
    Type: 'AWS::Neptune::DBInstance'
    Properties:
      DBClusterIdentifier:
        Ref: NeptuneDBCluster
      DBInstanceClass:
        Ref: DbInstanceType
    DependsOn:
      - NeptuneDBCluster
Outputs:
  DBClusterId:
    Description: Neptune Cluster Identifier
    Value:
      Ref: NeptuneDBCluster

• For a custom cluster parameter group, update the DBClusterParameterGroup family to the one corresponding to new engine version, namely neptune1.2.

• For a default cluster parameter group, update the DBClusterParameterGroup to the default corresponding to new engine version, namely default.neptune1.2.

• For each DBInstance attached to the DBCluster, update a default DBParameterGroup to the one in the family used by new engine version (here default.neptune1.2), and a custom parameter group to one that uses the family supported by the new engine version (here neptune1.2).

• Set the DBInstanceParameterGroupName property to the parameter group in the family supported by the new engine version.

The template should look like this:

Description: Template to update Neptune Stack to Engine Version 1.2.0.1 using custom and default Parameter Groups 
Parameters:
  DbInstanceType:
    Description: Neptune DB instance type
    Type: String
    Default: db.r5.large
Resources:
  NeptuneDBClusterParameterGroup:
    Type: 'AWS::Neptune::DBClusterParameterGroup'
    Properties:
      Family: neptune1.2
      Description: 'NeptuneDBClusterParameterGroup with family neptune1.2'
      Parameters:
        neptune_enable_audit_log: 0
  NeptuneDBParameterGroup:
    Type: 'AWS::Neptune::DBParameterGroup'
    Properties:
      Family: neptune1.2
      Description: 'NeptuneDBParameterGroup1 with family neptune1.2'
      Parameters:
        neptune_query_timeout: 20000
  NeptuneDBCluster:
    Type: 'AWS::Neptune::DBCluster'
    Properties:
      EngineVersion: 1.2.0.2
      DBClusterParameterGroupName:
        Ref: NeptuneDBClusterParameterGroup
      DBInstanceParameterGroupName: default.neptune1.2
    DependsOn:
      - NeptuneDBClusterParameterGroup
  CustomNeptuneDBInstance:
    Type: 'AWS::Neptune::DBInstance'
    Properties:
      DBClusterIdentifier:
        Ref: NeptuneDBCluster
      DBInstanceClass:
        Ref: DbInstanceType
      DBParameterGroupName:
        Ref: NeptuneDBParameterGroup
    DependsOn:
      - NeptuneDBCluster
      - NeptuneDBParameterGroup
  DefaultNeptuneDBInstance:
    Type: 'AWS::Neptune::DBInstance'
    Properties:
      DBClusterIdentifier:
        Ref: NeptuneDBCluster
      DBInstanceClass:
        Ref: DbInstanceType
      DBParameterGroupName: default.neptune1.2
    DependsOn:
      - NeptuneDBCluster
Outputs:
  DBClusterId:
    Description: Neptune Cluster Identifier
    Value:
      Ref: NeptuneDBCluster

Now use AWS CloudFormation to run the revised template.