Avoid joining to the same table multiple times
Try how good this works for you

Converted left join to one inner join for table Two for both t_id's. And on computing min, using case statement to split to two columns based on the value t_id. What we save here is scanning the table two time and an inner join compared to left join

SELECT
    o.p_id,
    o.k_id,
    MIN(case when t.t_id = 1 then t.pos else null end) AS t_pos,
    MIN(case when t.t_id = 2 then t.pos else null end) AS t1_pos
FROM
    one o
INNER JOIN two T ON t.p_id = o.p_id
AND t.k_id = o.k_id
AND t.t_id =any ( 1,2)
WHERE
    o.p_id = 1
AND o.c_id = 1
GROUP BY
    1,
    2
LIMIT 1