This guy is a rock, would you have played this more aggressively?

Party Poker No-Limit Hold'em, $ BB (6 max, 5 handed)

Hero ($51.10)
BB ($48.70)
UTG ($66.10)
MP ($42.95)
Button ($42.10)

Preflop: Hero is SB with , . Hero posts a blind of $0.25.
3 folds, Hero (poster) completes, BB checks.

Flop: ($1) , , (2 players)
Hero bets $1, BB raises to $3, Hero calls $2.

Turn: ($7) (2 players)
Hero checks, BB bets $4, Hero calls $4.

River: ($15) (2 players)
Hero checks, BB checks.

Final Pot: $15

Results in white below:
Hero has 3s 6s (one pair, fives).
BB has Ah 8h (two pair, eights and fives).
Outcome: BB wins $15.

I would have folded the turn. You could be drawing dead to a lot of hands.

As soon as that board pairs I am out. The time to make your play at that hand would have been to raise on the flop.

I am sorry I should have added more to it. The combined possibility of you drawing dead to many hands and the board pairing in this situation. I would fold. I do not chase a flush on a paired board ever. No exceptions. I have been burned alot earlier in my days.

I'll chase a flush on a paired board, but only under certain circumstances.

I'm NOT chasing when the pair is on the flop.

I'm NOT chasing when the pair could easily boat someone because they're connected.
For example. 854 and the turn is a 5--NO because 54 is a very likely hand. K 8 3 and the turn is another 3. Yes because K3 and 83 are unlikley holdings.