There are probably other options, but I've used the roads I am most familiar with and that I would ride (some, admittedly, only if I had to, but they are viable). Here's what I came up with:

From Fountainbleu, take the Tammany Trace west past Abita Springs.

Fourth right off of the Trace after the Brew Pub is Schoen Road. There are several sprayed markers on this road as well since it is a popular local route.

Cross Hwy 36 to the left to the catty-cornered Plantation Street.
Options from this point:
-Take a right onto Lowe Davis Road, which is very smooth and a shorter route, but there is no shoulder and depending on the time of day car traffic can be steady.
-Continue on Plantation past Lowe Davis and take the next right on Shady Lane. This is a quite subdivision with minimal traffic, though it is a bit longer. The road dead ends on Lowe Davis anyway, and then just take a left onto it (just cuts down time on what can be a busier road.

Options from this point:
-Take a left onto Hwy 59 up to Hwy 21, take a left on 21 and then take the next right onto 1082 to head NNW. Disclaimer: I’ve never ridden on 59, traffic tends to be fairly fast, and the shoulder is not good. Hwy 21, however, has good shoulders and Hwy 1082 is a very popular riding option around here (and has nice scenery).
-Usual route: cross Hwy 59 to continue on Lowe Davis for a few miles (not more than 2 I think) until you come to Hwy 1083 (you’ll see open fields and a couple of large barns on the left as you come to it). Take a left onto the Hwy 1083 and continue on NNW. Congratulations, there are some “hills” from this point.
Hwy 1083 is bisected by Hwy 21, with 1083 North continuing once you take 21 around a curve to the right. There is a Backroads Mercantile at the T intersection of 21 and 1083 if you need to refill/stock up on snacks. If you stop during the weekday, Pete the owner is very nice.

Either option of Hwy 1082 or 1083 dead end at Hwy 40. This is a curvy road with minimal shoulders. Road bikers are not unheard of on this road, but sometimes drivers like to go fast, so operate with some extra caution.

OK, this is the end of my usual commute roads and the roads I am most familiar with by bicycle. If you/your guest(s) are looking for a scenic route I would choose to take either 1082 or 1083 up to 40.

One option to get to the Southern Tier is to take a left on 40 (from either 1082 or 1083) and take the Isabel Swamp road (2 miles or so of gravel and a bridge over the Bogue Chitto river, lightly traveled but rustic to put it mildly. If this is a fully loaded touring rig, this would probably not be the best option, but I'm looking forward to riding this on the 28 GravelKings I just put on one of my bikes) to Hwy 16. Take a left on 16 (again, no shoulders, but decent pavement) and then take a right on 1073 and then a right on Hwy 60. This will lead back to the portion of Hwy 60 that is part of the Southern Tier route.

Direct route option if scenery is not the major factor at this point:
Either take a right on Hwy 21 and bypass 1082 and 1083 or if taking either one and riding on 40, take a left back on 21 when 40 ends at the Sixth Ward Middle School. Follow 21 (wide but rough shoulders, high traffic road but I’ve ridden it many times with no issues). After the middle school, you pass a Dollar General on the left and Texaco and Shell stations on the right (along with the Bush House of Seafood immediately after the Shell, which is surprisingly good food but I think it’s closed for the season). Hwy 21 comes to a “T” shortly after this point, so hang a left and continue north.

Disclaimer:
I have not ridden this next bit by bicycle, only driven it. 21 still has decent shoulders, but there is a higher speed limit on this stretch.
Continue north on 21 up until 1075 on the left. Take that and continue until it dead ends on Hwy 60/Southern Tier. You can continue on 21 and ride into Bogalusa if you choose, but I even dislike driving this as there is not much to see.
Also, Bogalusa is home to a large paper mill, so there is a definite odor in the area.

Rough estimate of distance from the Park to Bogalusa would be 30 miles? Taking 1082->1073->60 would probably be the longest way, and could add 10-12 miles? I wanted to add a map with the routes highlighted, but Google Earth is giving me issues tonight, sorry about that.

Please feel free to pass along any questions or concerns your guest may have. I'm happy to help and I certainly understand if this needs some clarification.