A Noetherian local ring
(
R
,
m
)
(R,\frak {m})
is called Buchsbaum if the difference
ℓ
(
R
/
q
)
−
e
(
q
,
R
)
\ell (R/\mathfrak {q})-e(\mathfrak {q}, R)
, where
q
\mathfrak {q}
is an ideal generated by a system of parameters, is a constant independent of
q
\mathfrak {q}
. In this article, we study the tight closure analog of this condition. We prove that in an unmixed excellent local ring
(
R
,
m
)
(R,\frak {m})
of prime characteristic
p
>
0
p>0
and dimension at least one, the difference
e
(
q
,
R
)
−
ℓ
(
R
/
q
∗
)
e(\mathfrak {q}, R)-\ell (R/\mathfrak {q}^*)
is independent of
q
\mathfrak {q}
if and only if the parameter test ideal
τ
p
a
r
(
R
)
\tau _{\mathrm {par}}(R)
contains
m
\frak {m}
. We also provide a characterization of this condition via derived category which is analogous to Schenzel’s criterion for Buchsbaum rings.