You should avoid using nested selects and I would go as far to say you should never use them in the actual select part of your statement. You will be running that select for each row that is returned. This is a really expensive operation. Rather use joins. It is much more readable and the performance is much better.
In your case the query below should help. Note the cases statement is still there, but now it is a simple compare operation.
select
p.product_id,
p.type_id,
p.product_name,
p.type,
case p.type_id when 10 then (CONCAT_WS(' ' , first_name, middle_name, last_name )) else (null) end artistC
from
Product p
inner join Product_Type pt on
pt.type_id = p.type_id
left join Product_ArtistAuthor paa on
paa.artist_id = p.artist_id
where
p.product_id = $pid
I used a left join since I don't know the business logic.