Need to create combinations of Parents/Children/Dogs, but can't get past 2nd level of CTE

Question

I've hit a brick wall and please ask help.

I'm building a table up for (in this example) a ticketing system where e.g. Ticket 1 will allow Parent: Either parent Peter or Sally, Child: to come without a child or with a child (either Sam, Pete or John), Dog: and either without a dog or with one of the dogs (DogMatix,Snowy,Bliksem)

The table layout is that the Parent/Child/Dog is a parameter, and I can get (with CTE) to the Parent/Child relationships to all show, but not the next level showing the dogs too

declare @Ticket table
(
TID int
)

declare @Param table
(PID int IDENTITY(1,1) not null,
TID int,
ParamType varchar(20),
Val  varchar(40)
)


Insert into @Ticket(TID) Select(1)
Insert into @Ticket(TID) Select(2)


Insert into @Param(TID,ParamType,Val) values(1,'Parent','ParentPeter')
Insert into @Param(TID,ParamType,Val) values(1,'Parent','ParentSally')
Insert into @Param(TID,ParamType,Val) values(1,'Child','ChildSam')
Insert into @Param(TID,ParamType,Val) values(1,'Child','ChildPete')
Insert into @Param(TID,ParamType,Val) values(1,'Child','ChildJohn')
Insert into @Param(TID,ParamType,Val) values(1,'Dog','DogDogMatix')
Insert into @Param(TID,ParamType,Val) values(1,'Dog','DogSnowy')
Insert into @Param(TID,ParamType,Val) values(1,'Dog','DogBliksem')

Insert into @Param(TID,ParamType,Val) values(2,'Parent','ParentParker')
Insert into @Param(TID,ParamType,Val) values(2,'Parent','ParentScarlett')
Insert into @Param(TID,ParamType,Val) values(2,'Child','ChildBrett')
Insert into @Param(TID,ParamType,Val) values(2,'Child','ChildBritney')
Insert into @Param(TID,ParamType,Val) values(2,'Child','ChildBronwyn')
Insert into @Param(TID,ParamType,Val) values(2,'Dog','DogRover')
Insert into @Param(TID,ParamType,Val) values(2,'Dog','Dog2')
Insert into @Param(TID,ParamType,Val) values(2,'Dog','Dog3')




;with cte  as (
Select TID,Val as 'Parent' 
from @Param p
where  p.ParamType = 'Parent'
 )
Select cte.TID,cte.Parent,p.Val as 'Child' 
from cte  
left outer join @Param p on p.TID = cte.TID and  p.ParamType = 'Child'




--The required results will be (for ticket 1)
--TID       Parent          Child           Dog
--1     ParentPeter     ChildSam        DogDogMatix
--1     ParentPeter     ChildSam        DogSnowy
--1     ParentPeter     ChildSam        DogBliksem
--1     ParentPeter     ChildPete       DogDogMatix
--1     ParentPeter     ChildPete       DogSnowy
--1     ParentPeter     ChildPete       DogBliksem
--1     ParentPeter     ChildJohn       DogDogMatix
--1     ParentPeter     ChildJohn       DogSnowy
--1     ParentPeter     ChildJohn       DogBliksem
--1     ParentPeter     null            null
--1     ParentPeter     ChildSam        null
--1     ParentPeter     ChildPete       null
--1     ParentPeter     ChildJohn       null

--1     ParentSally     ChildSam        DogDogMatix
--1     ParentSally     ChildSam        DogSnowy
--1     ParentSally     ChildSam        DogBliksem
--1     ParentSally     ChildPete       DogDogMatix
--1     ParentSally     ChildPete       DogSnowy
--1     ParentSally     ChildPete       DogBliksem
--1     ParentSally     ChildJohn       DogDogMatix
--1     ParentSally     ChildJohn       DogSnowy
--1     ParentSally     ChildJohn       DogBliksem
--1     ParentSally     null            null
--1     ParentSally     ChildSam        null
--1     ParentSally     ChildPete       null
--1     ParentSally     ChildJohn       null