It would help if you show us the error that pops out.
On my machine, after some massaging of your code, the error seems to be that OrderedLogisitc does not have a CDF, which is necessary for the copula object to provide a pdf (and therefore it wont work).
Also, the in this version, the Y2[i] ~ OrderedLogistic(Y2st[i]+μ2[i],cutpoints) line does nothing since it is not used later, right ? So you might just remove it.
Moreover, you may change
# Regression parameters
μ1 = zeros(typeof(θ₁), n)
μ2 = zeros(typeof(θ₂), n)
ρ = zeros(typeof(θ), n)
Y2st ~ filldist(Logistic(0, 1.), n)
# Implement Shrinkage Priors
for i in 1:n
μ1[i] = θ₁ .+ sum(X[i,:].* β1)
μ2[i] = θ₂ .+ sum(X[i,:].* β2)
ρ[i] = exp(θ .+ sum(X[i,:].* β))
Y2[i] ~ OrderedLogistic(Y2st[i]+μ2[i],cutpoints)
#Now the copula is between Y1 (observed) and Y2st (latent)
Turing.Turing.@addlogprob! loglikelihood(SklarDist(SurvivalCopula(ClaytonCopula(2,ρ[i]),(1,2)), (Normal(μ1[i], 1.), Logistic(μ2[i],1.))), [Y1[i],Y2st[i]+μ2[i]] )
end
to simply
# Regression parameters
# nothing here.
# Implement Shrinkage Priors
for i in 1:n
μ1i = θ₁ .+ sum(X[i,:].* β1)
μ2i = θ₂ .+ sum(X[i,:].* β2)
ρi = exp(θ .+ sum(X[i,:].* β))
#Now the copula is between Y1 (observed) and Y2st (latent)
Turing.Turing.@addlogprob! loglikelihood(SklarDist(SurvivalCopula(ClaytonCopula(2,ρi),(1,2)), (Normal(μ1i, 1.), Logistic(μ2i,1.))), [Y1[i],Y2st[i]+μ2[i]] )
end
Removing useless vectors will probably make your function a bit faster.